Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom fo/u fopkj Hkh# tu] ugha vkjehks dke] foif s[k NksMs qja e/;e eu dj ';kea iq#"k flag ladyi dj] lgs foif vusd] ^cuk^ u NksMs /;s; dks] j?kqcj jk[ks VsdAA jfp% ekuo /kez izksk l~xq# Jh jknksmklth egkjkt STUDY PACKAGE Subject : Mathematics Topic : Sequece & Pogessio Available Olie : wwwmathsbysuhagcom Idex Theoy Shot Revisio Execise (Ex 6) Assetio & Reaso Que fom Compt Exams 6 9 Ys Que fom IIT-JEE(Advaced) 7 Ys Que fom AIEEE (JEE Mai) Studet s Name : Class Roll No : : Addess : Plot No 7, III- Floo, Nea Patida Studio, Above Bod Classes, Zoe-, MP NAGAR, Bhopal : 0 90 90 7779, 9890 888, WhatsApp 9009 60 9 wwwtekoclassescom wwwmathsbysuhagcom R Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom Sequece & Pogessio Sequece : A sequece is a fuctio whose domai is the set N of atual umbes Sice the domai fo evey sequece is the set N of atual umbes, theefoe a sequece is epeseted by its age If f : N R, the f() t N is called a sequece ad is deoted by {f(), f(), f(),} {t, t, t, } {t } Real Sequece : A sequece whose age is a subset of R is called a eal sequece Examples : (i),, 8,, (ii),,,, (iii), 9, 7, 8, Types of Sequece : O the basis of the umbe of tems thee ae two types of sequece (i) Fiite sequeces : A sequece is said to be fiite if it has fiite umbe of tems (ii) Ifiite sequeces : A sequeceis said to be ifiite if it has ifiite umbe of tems Solved Example # Wite dow the sequece whose th tem is ( ) (i) (ii) Solutio (i) Let t Seies put,,,, we get t, t, t 8, t so the sequece is,, 8,, ( ) (ii) Let t put,,,, so the sequece is,,,, 9 7 8 By addig o substactig the tems of a sequece, we get a expessio which is called a seies a, a, a,a is a sequece, the the expessio a a a a is a seies Example (i) (ii) 8 6 Pogessio : It is ot ecessay that the tems of a sequece always follow a cetai patte o they ae descibed by some explicit fomula fo the th tem Those sequeces whose tems follow cetai pattes ae called pogessios A aithmetic pogessio (AP): AP is a sequece whose tems icease o decease by a fixed umbe This fixed umbe is called the commo diffeece If a is the fist tem & d the commo diffeece, the AP ca be witte as a, a d, a d, a ( ) d, Example,,, (i) th tem of a AP Let a be the fist tem ad d be the commo diffeece of a AP, the t a ( ) d whee d a a Solved Example # If t of a AP is 6 ad t 6, fid t 0 Solutio Let a be the fist tem ad d be the commo diffeece so t a d 6 (i) ad t a d 6 (ii) equatio (i) (ii) 0d d a so t 0 9 9 Solved Example # Fid the umbe of tems i the sequece,, 0, 08 Solutio a, d 8 so 08 ( )8 (ii) The sum of fist tems of ae AP If a is fist tem ad d is commo diffeece the S [a ( ) d] [a ] t, whee is the last tem ad t is the middle tem (iii) th tem of a AP whe sum of fist tems is give is t s S Solved Example # Fid the sum of all atual umbes divisible by, but less tha 00 Solutio All those umbes ae, 0,, 0, 9 If Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 9 Hee a 9 9 so S ( 9) 90 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom Solved Example # Fid the sum of all the thee digit atual umbes which o divisio by 7 leaves emaide Solutio All these umbes ae 0, 08,, 997, to fid 997 0 ( ) 7 9 FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom 9 so S [0 997] 708 7 Solved Example # 6 The sum of tems of two APs ae i atio Fid the atio of thei th tems 7 Sol Let a ad a be the fist tems ad d ad d be the commo diffeeces of two APs espectively the [a ( )d ] a d 7 7 [a ( )d ] 7 7 a d Fo atio of th tems 0 7() so atio of th tems is () 7 8 Solved Example # 7 If sum of tems of a sequece is give by S, fid its 0 th tem Solutio Let t is th tem of the sequece so t s s ( ) ( ) so t 0 0 Self Pactice Poblems : Which tem of the sequece 00, 000, 99, 990, 98, cotais the fist egative tem As 0 Fo a AP show that t m t m t m Fid the maximum sum of the AP 0, 8, 6,,, As 0 Popeties of AP (i) The commo diffeece ca be zeo, positive o egative (ii) If a, b, c ae i AP b a c & if a, b, c, d ae i AP a d b c (iii) (iv) (v) Thee umbes i AP ca be take as a d, a, a d; fou umbes i AP ca be take as a d, a d, a d, a d; five umbes i AP ae a d, a d, a, a d, a d & six tems i AP ae a d, a d, a d, a d, a d, a d etc The sum of the tems of a AP equidistat fom the begiig & ed is costat ad equal to the sum of fist & last tems Ay tem of a AP (except the fist) is equal to half the sum of tems which ae equidistat fom it a / (a k a k ), k < Fo k, a (/) (a a ); Fo k, a (/) (a a ) ad so o (vi) If each tem of a AP is iceased, deceased, multiplied o divided by the same o zeo umbe, the the esultig sequece is also a AP Solved Example # 8 The sum of thee umbes i AP is 7 ad the sum of thei squaes is 9, fid them Solutio Let the umbes be a d, a, a d so a 7 a 9 Also (a d) a (a d) 9 a d 9 d d ± theefoe umbes ae, 9, Solved Example # 9 If a, a, a, a, a ae i AP with commo diffeece 0, the fid the value of whe a Solutio As a, a, a, a, a, ae i AP, we have a a a a a Hece a i 0 i Solved Example # 0 If b c, c a, ae i AP pove that a a b, b, c ae also i AP Solutio b c, c a, ae i AP a b b c c a c a a b c a b c a b c a (c a)(b c) (a b)(c a) b a c b b b c a b a c b a, b, c ae i AP i a i Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom b c a c a b a b c Solved Example # If,, ae i AP, the,, ae also i AP a b c a b c b c a c a b a b c Solutio Give,, ae i AP a b c Add to each tem b c a c a b a b c,, ae i AP a b c divide each by a b c,, ae i AP a b c Aithmetic Mea (Mea o Aveage) (AM): If thee tems ae i AP the the middle tem is called the AM betwee the othe two, so if a, b, c ae i AP, b is AM of a & c (a) Aithmetic Meas Betwee Two Numbes: If a, b ae ay two give umbes & a, A, A,, A, b ae i AP the A, A, A ae the AM s betwee a & b A a b a, A a ( b a),, A a ( b a ) NOTE : Sum of AM s iseted betwee a & b is equal to times the sigle AM betwee a & b ie A A whee A is the sigle AM betwee a & b Solved Example # Betwee two umbes whose sum is, a eve umbe of AMs is iseted, the 6 sum of these meas exceeds thei umbe by uity Fid the umbe of meas Solutio Let a ad b be two umbes ad AMs ae iseted betwee a ad b the (a b) give a b 6 6 6 Numbe of meas Solved Example # Iset 0 AM betwee ad 86 Solutio Hee is the fist tem ad 86 is the d tem of AP so 86 ()d d so the seies is, 6, 0,,, 8, 86 equied meas ae 6, 0,,8 Self Pactice Poblems : If AM betwee p th ad q th tems of a AP be equal to the AM betwee th ad s th tem of the AP the pove that p q s If AMs ae iseted betwee 0 ad 80 such that fist meas : last mea :, fid As a b 6 Fo what value of,, a b is the AM of a ad b As 0 a b Geometic Pogessio (GP) GP is a sequece of umbes whose fist tem is o zeo & each of the succeedig tems is equal to the poceedig tems multiplied by a costat Thus i a GP the atio of successive tems is costat This costat facto is called the commo atio of the seies & is obtaied by dividig ay tem by that which immediately poceeds it Theefoe a, a, a, a, a, is a GP with a as the fist tem & as commo atio Example,, 8, 6 Example,,, 9 7 8 (i) th tem a a (ii) Sum of the fist tems ie S, a, (iii) Sum of a ifiite GP whe < Whe a 0 if < theefoe,s Solved Example # : If the fist tem of GP is 7, its th tem is 8 ad sum of fist tems is 889, the fid the fifth tem of GP Solutio Give a 7 the fist tem t a 7() 8 7 8 a( ) 7( Also S ) 8 7 889 Hece T a 7() Solved Example # : The fist tem of a ifiite GP is ad ay tem is equal to the sum of all the succeedig tems Fid the seies Solutio Let the GP be,,,, Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom give coditio, Hece seies is,,, 8, FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Solved Example # 6:Let S 8 fid the sum of (i) fist 0 tems of the seies (ii) ifiite tems of the seies 0 0 Solutio (i) S 0 9 (ii) S Self Pactice Poblems : Fid the GP if the commo atio of GP is, th tem is 86 ad sum of fist tems is 78 As, 6, 8,, 6, 86 If the p th, q th, th tems of a GP be a, b, c espectively, pove that a q b p c p q A GP cosist of tems If the sum of the tems occupyig the odd places is S ad that of the tems S occupyig the eve places is S the fid the commo atio of the pogessio As S The sum of ifiite umbe of tems of a GP is, ad the sum of thei cubes is 9, fid the seies As 6,,, Popeties of GP (i) If a, b, c ae i GP b ac, i geeal if a, a, a, a, a, a ae i GP, the a a a a a a (ii) Ay thee cosecutive tems of a GP ca be take as a, a, a, i geeal we take a a k, k, a k,a, a, a,a k i case we have to take k tems i a GP a a (iii) Ay fou cosecutive tems of a GP ca be take as,, a, a, i geeal we take a k, a a k,, a, a k i case we have to take k tems i a GP (iv) If each tem of a GP be multiplied o divided o aised to powe by the some ozeo quatity, the esultig sequece is also a GP (v) If a, a, a, ad b, b, b, ae two GP s with commo atio ad espectively the the sequece a b, a b, a b, is also a GP with commo atio (vi) If a, a, a,ae i GP whee each a i > 0, the log a, loga, loga,ae i AP ad its covese is also tue Solved Example # 7: Fid thee umbes i GP havig sum 9 ad poduct 6 a Solutio Let the thee umbes be, a, a so a 9 (i) ad a 6 a 6 so fom (i) 6 6 0, Hece the thee umbes ae, 6, 9 Solved Example # 8: Fid the poduct of tems i GP whose 6 th is Solutio: Usig the popety a a a a 0 a a 9 a 6 Hece poduct of tems p Solved Example # 9:Usig GP expess 0 ad as fom q Solutio Let x 0 0 0 00 000 0000 0 00 000 0000 0 0 9 Let y 00 000 0000 0 0 0 Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom 0 7 0 0 0 Solved Example # 0 Evaluate 7 77 777 upto tems Solutio Let S 7 77 777 upto tems 9 7 [9 99 999 ] 9 7 [(0 ) (0 ) (0 ) upto tems] 9 7 [0 0 0 0 ] 9 7 (0 ) 0 9 7 [0 9 0] 8 Geometic Meas (Mea Popotioal) (GM): If a, b, c ae i GP, b is the GM betwee a & c b² ac, theefoe b a c ; a > 0, c > 0 (a) Geometic Meas Betwee a, b: If a, b ae two give umbes & a, G, G,, G, b ae i GP The G, G, G,, G ae GMs betwee a & b G a(b/a) /, G a(b/a) /,, G a(b/a) / NOTE : The poduct of GMs betwee a & b is equal to the th powe of the sigle GM betwee a & b ie G (G) whee G is the sigle GM betwee a & b Solved Example # Iset GMs betwee ad 86 b Solutio Commo atio of the seies is give by () / a Hece fou GMs ae 6, 8,, 6 Self Pactice Poblems : The sum of thee umbes i GP i 70, if the two extemes be multiplied each by ad the mea by, the poducts ae i AP Fid the umbes As 0, 0, 0 If a, b 0 0 0 0 ad c 0 0 0 0 0, the pove that (i) a is a composite umbe (ii) a bc Hamoic Pogessio (HP) : A sequece is said to HP if the ecipocals of its tems ae i AP If the sequece a, a, a,, a is a HP the /a, /a,, /a is a AP & covese Hee we do ot have the fomula fo the sum of the tems of a HP Fo HP whose fist tem is a ad secod tem is b, the th a b tem is t If a, b, c ae i HP b ac o a b ( )( a b) a c c a b b c a b a a b a NOTE : (i) If a, b, c ae i AP (ii) If a, b, c ae i GP a b b c Hamoic Mea (HM): If a, b, c ae i HP, b is the HM betwee a & c, the b ac/[a c] If a, a, a ae o-zeo umbes the HM H of these umbes is give by H a a a Solved Example # : If m th tem of HP is, while th tem is m, fid its (m ) th tem Solutio: Give T m o ; whee a is the fist tem ad d is the commo diffeece of the coespodig AP so a (m )d a (m ) d m ad a ( ) d (m )d o d m m m ( m ) so a m m m m Hece T (m ) a (m d) d m m Solved Example # : Iset HM betwee / ad / Solutio Let d be the commo diffeece of coespodig AP so d H o H Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't b c Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 6 of 6
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom 7 H 9 H o H 7 o H 9 H o H Solved Example # : If p th, q th, th tems of a HP be a, b, c espectively, pove that (q )bc ( p) ac (p q) ab 0 Solutio Let x be the fist tem ad d be the commo diffeece of the coespodig AP so x (p )d a (i) x (q ) d b (ii) x ( ) d (iii) c (i) - (ii) ab(p q)d b a (iv) (ii) - (iii) bc (q )d c b (v) (iii) - (i) ac ( p) d a c (vi) (iv) (v) (vi) gives bc (q ) ac( p) ab (p q) 0 Self Pactice Poblems : If a, b, c be i HP, show that a : a b a c : a c If the HM betwee two quatities is to thei GMs as to, pove that the quatities ae i atio to 9 H If H be the hamoic mea of a ad b the fid the value of a If a, b, c, d ae i HP, the show that ab bc cd ad Relatio betwee meas : (i) H As 0 b If A, G, H ae espectively AM, GM, HM betwee a & b both beig uequal & positive the, G² AH ie A, G, H ae i GP Solved Example # :The AM of two umbes exceeds the GM by ad the GM exceeds the HM by 6 ; fid the umbes Solutio Let the umbes be a ad b, ow usig the elatio G AH 6 G G 9 G G G 6 0 (ii) ie ab 6 also a b Hece the two umbes ae ad AM GM HM Let a, a, a, a be positive eal umbes, the we defie thei a a a a AM, thei GM (a a a a ) / ad thei HM It ca be show that a a a AM GM HM ad equality holds at eithe places iff a a a a Solved Example # 6 If a, b, c, > 0 pove that b a c b a c Solutio Usig the elatio AM GM we have a b c b c a a b c a b c b c a b c a Solved Example # 7 Fo o-zeo x, y, z pove that (x y z) x y z 9 Solutio Usig the elatio AM HM x y z x y z (x y z) x y z 9 Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 7 of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom Sol Ex # 8: If a i > 0 i N such that a Solutio Usig AM GM a a a a i i a a As a a a a Hece ( a ) ( a ) ( a ), the pove that ( a ) ( a ) ( a ) ( a ) ( a ) ( a ) ( a ) / Solved Example # 9 If > 0 pove that > Solutio Usig the elatio AM GM o the umbes,,, we have > ( ) / Equality does ot hold as all the umbes ae ot equal () > () () > > Sol Ex # 0 Fid the geatest value of xyz fo positive value of x, y, z subject to the coditio xy yz zx Solutio Usig the elatio AM GM xy yz zx (x y z ) / (x y z) / xyz 8 Solved Example # If a, b, c ae i HP ad they ae distict ad positive the pove that a c > b Solutio Let a ad c be two umbes a c the > (a c ) / a c > (ac) / (i) Also GM > HM ie ac > b (ac) / > b (ii) hece fom (i) ad (ii) a c > b Self Pactice Poblems : If a, b, c ae eal ad distict the show that a ( b ) b ( c ) c ( a ) > 6abc Pove that > ( ) If a, b, c, d be fou distict positive quatities i GP the show that (i) a d > b c (ii) > ab cd bd ac ad Pove that ABC is a equilateal tiagle iff ta A ta B ta C If a, b, c > 0 pove that [( a) ( b) ( c)] 7 > 7 7 a b c Aithmetico-Geometic Seies: A seies each tem of which is fomed by multiplyig the coespodig tem of a AP & GP is called the AitHMeticoGeometic Seies eg x x 7x Hee,,, ae i AP &, x, x, x ae i GP Sum of tems of a AithmeticoGeometic Seies: Let S a (a d) (a d) ² [a ( )d] a d the S a ( ) d, Sum To Ifiity: If < & the Limit 0 S Solved Example # Fid the sum of the seies 7 0 to tems 7 0 Solutio Let S 7 S (i) (ii) S S a a a d (i) (ii) a a Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 8 of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom 7 7 ( 7) S 6 6 Solved Example # : Evaluate x x x upto ifiity whee x < Solutio Let S x x x (i) xs x x x (ii) (i) - (ii) ( x) S x x x o S ( x) Solved Example # 6 Evaluate ( b) ( b b ) to ifiite tems fo b < Solutio Let S ( b) ( b b ) (i) S ( b) (ii) (i) - (ii) ( )S b b b S ( b)( ) Self Pactice Poblems : Evaluate 00 00 As 99 0 Evaluate x 6x 0x upto ifiite tem whee x < As ( x) Sum to tems of the seies Impotat Results (i) (iii) (v) (vii) (a ± b ) a ± b (ii) Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't As k k k k times k; whee k is a costat(iv) ² ( ) ( ) 6 (vi) a i a j (a a a ) (a a a ) i j k a k a ( ) Solved Example # 7: Fid the sum of the seies to tems whose geeal tem is Solutio S T ( ) ( ) Solved Example # 8:T k k k the fid k Solutio T k Solved Example # 9: Solutio: k o ( ) k k ( ) ( ) 6 i i j k j k k ( ) T k ( ) ( ) 6 Fid the value of the expessio i i i i j i i j i (i ) i i i j k j ( ) ( ) ( ) ( ) 6 ( ) ( ) ( ) [ ] 6 METHOD OF DIFFERENCE Type Let u, u, u be a sequece, such that u u, u u, is eithe a AP o a GP the th tem u of this sequece is obtaied as follows S u u u u (i) S u u u u (ii) (i) (ii) u u (u u ) (u u ) (u u ) Whee the seies (u u ) (u u ) (u u ) is Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 9 of 6
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom eithe i AP o i GP the we ca fid u ad hece sum of this seies as S Solved Example # 0 Fid the sum to -tems 7 Solutio Let S 7 T (i) S 7 T T (ii) (i) (ii) T 6 8 (T T ) [8 ( )] ( ) ( ) Hece S ( ) ( )( ) ( ) ( ) 6 Solved Example # Fid the sum to -tems 0 Solutio Let S 0 T (i) S 0 T T (ii) (i) (ii) T ( 6 T T ) So T T S T Type If possible expess th tem as diffeece of two tems as t f() f( ± ) This ca be explaied with the help of examples give below Solved Example # Fid the sum to -tems of the seies Solutio Let T be the geeal tem of the seies So T ( ) To expess t f() f( ) multiply ad divide t by [( ) ( )] so T ( ) [( ) ( )] Let f() ( ) ( ) [ ( ) ( ) ( ) ( )] so T [f() f( )] Now S T T T T T T [ 0], T [ ], T [ ] T [() ( ) ( ) ( )] S ( ) ( ) Hece sum of seies is f() f(0) Solved Example # Sum to tems of the seies ( x)( x) ( x)( x) ( x)( x) Solutio Let T be the geeal tem of the seies [ ( )x] ( x) T So T ( x)( ( )x) x ( x)( ( )x) x x ( ) x T f() f( ) S T T T T T x x ( ) x ( x)[ ( )x] 6 Solved Example # Su to tems of the seies Solutio Let T ( )( ) ( )( ) S ( )( ) ( )( ) k u ( ) ( )( ) Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 0 of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom ( ) ( )( ) [ ] Note : It is ot always ecessay that the seies of fist ode of diffeeces ie u u, u u u u, is always eithe i AP o i GP i such case let u T, u u T, u u T, u u T So u T T T (i) u T T T T (ii) (i) (ii) T T (T T ) (T T ) (T T ) Now, the seies (T T ) (T T ) (T T ) is seies of secod ode of diffeeces ad whe it is eithe i AP o i GP, the u u T Othewise i the simila way we fid seies of highe ode of diffeeces ad the th tem of the seies With the help of followig example this ca be explaied Solved Example # Fid the th tem ad the sum of tem of the seies,, 6, 80, 0, Solutio Let S 6 80 0 T (i) S 6 80 0 T T (ii) (i) (ii) T 0 70 0 (T T ) (iii) T 0 70 0 (T T ) (T T ) (iv) (iii) (iv) T T 8 0 6 [ ( ) 6] [ ] T T geeal tem of give seies is T T Hece sum of this seies is S ( ) ( )( ) ( ) ( 6 7 ) ( ) ( ) ( ) Solved Example # 6: Fid the geeal tem ad sum of tems of the seies 9, 6, 9,, 0 Sol Let S 9 6 9 0 T (i) S 9 6 9 0 T T (ii) (i) (ii) T 9 7 9 (T T ) (iii) T 9 7 9 (T T ) (T T ) (iv) (iii) (iv) T T 9 ( ) 6 7 6 [ ] 6() ( ) tems Geeal tem is T 6() Also sum S T 6 ( ) ( ) 6 6( ) Self Pactice Poblems : Sum to tems the followig seies (i) As (ii) 7 7 9 As (iii) 9 6 0 7 As (iv) 0 8 As ( ) (v) 86 As ( ) ( )( ) ( ) ( 8) ( 9) Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom SHORT REVESION (SEQUENCES AND SERIES) DEFINITION : A sequece is a set of tems i a defiite ode with a ule fo obtaiig the tems eg, /, /,, /, is a sequece AN ARITHMETIC PROGRESSION (AP) :AP is a sequece whose tems icease o decease by a fixed umbe This fixed umbe is called the commo diffeece If a is the fist tem & d the commo diffeece, the AP ca be witte as a, a d, a d, a ( )d, th tem of this AP t a ( )d, whee d a a - T h e s u m o f t h e f i s t t e m s o f t h e A P i s g i v e b y ; S [ a ( )d] [a l] whee l is the last tem NOTES :(i) If each tem of a AP is iceased, deceased, multiplied o divided by the same o zeo umbe, the the esultig sequece is also a AP (ii) Thee umbes i AP ca be take as a d, a, a d ; fou umbes i AP ca be take as a d, a d, a d, a d ; five umbes i AP ae a d, a d, a, a d, a d & six tems i AP ae a d, a d, a d, a d, a d, a d etc (iii) The commo diffeece ca be zeo, positive o egative (iv) The sum of the two tems of a AP equidistat fom the begiig & ed is costat ad equal to the sum of fist & last tems (v) Ay tem of a AP (except the fist) is equal to half the sum of tems which ae equidistat fom it (vi) t S S (vii) If a, b, c ae i AP b a c GEOMETRIC PROGRESSION (GP) : GP is a sequece of umbes whose fist tem is o zeo & each of the succeedig tems is equal to the poceedig tems multiplied by a costat Thus i a GP the atio of successive tems is costat This costat facto is called the COMMON RATIO of the seies & is obtaied by dividig ay tem by that which immediately poceeds it Theefoe a, a, a, a, a, is a GP with a as the fist tem & as commo atio (i) th tem a (ii) Sum of the I st a tems ie S, if (iii) Sum of a ifiite GP whe < whe a 0 if < theefoe,s ( ) (iv) If each tem of a GP be multiplied o divided by the same o-zeo quatity, the esultig sequece is also a GP (v) Ay cosecutive tems of a GP ca be take as a/, a, a ; ay cosecutive tems of a GP ca be take as a/, a/, a, a & so o (vi) If a, b, c ae i GP b ac HARMONIC PROGRESSION (HP) :A sequece is said to HP if the ecipocals of its tems ae i AP If the sequece a, a, a,, a is a HP the /a, /a,, /a is a AP & covese Hee we do ot have the fomula fo the sum of the tems of a HP Fo HP whose fist tem is a & secod tem is b, the th a b tem is t b ( )( a b) ac a If a, b, c ae i HP b o a c c a b b c MEANS ARITHMETIC MEAN :If thee tems ae i AP the the middle tem is called the AM betwee the othe two, so if a, b, c ae i AP, b is AM of a & c AM fo ay positive umbe a, a,, a is ; A a a a a - ARITHMETIC MEANS BETWEEN TWO NUMBERS : If a, b ae ay two give umbes & a, A, A,, A, b ae i AP the A, A, A ae the AM s betwee a & b A a b a, A a ( b a),, A a ( b a ) a d, a d,, A a d, whee d b a NOTE : Sum of AM s iseted betwee a & b is equal to times the sigle AM betwee a & b ie A A whee A is the sigle AM betwee a & b GEOMETRIC MEANS : If a, b, c ae i GP, b is the GM betwee a & c Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom b² ac, theefoe b a c ; a > 0, c > 0 -GEOMETRIC MEANS BETWEEN a, b : If a, b ae two give umbes & a, G, G,, G, b ae i GP The G, G, G,, G ae GMs betwee a & b G a(b/a) /, G a(b/a) /,, G a(b/a) / a, a², a, whee (b/a) / NOTE: The poduct of GMs betwee a & b is equal to the th powe of the sigle GM betwee a & b ie G (G) whee G is the sigle GM betwee a & b HARMONIC MEAN :If a, b, c ae i HP, b is the HM betwee a & c, the b ac/[a c] THEOREM : If A, G, H ae espectively AM, GM, HM betwee a & b both beig uequal & positive the, (i) G² AH (ii) A > G > H (G > 0) Note that A, G, H costitute a GP ARITHMETICO-GEOMETRIC SERIES : A seies each tem of which is fomed by multiplyig the coespodig tem of a AP & GP is called the Aithmetico-Geometic Seies eg x x 7x Hee,,, ae i AP &, x, x, x ae i GP Stadat appeaace of a Aithmetico-Geometic Seies is Let S a (a d) (a d) ² [a ( )d] SUM TO INFINITY : THEOREMS :(i) (iii) (i) (ii) (iii) (iv) If < & the Limit 0 S (a ± b ) SIGMA NOTATIONS k k ; whee k is a costat ( ) ² ( ) ( ) 6 ( ) a ± b (ii) RESULTS (sum of the fist atual os) k a k (sum of the squaes of the fist atual umbes) 0 ( ) ( ) (² ) (sum of the cubes of the fist atual umbes) a d METHOD OF DIFFERENCE : If T, T, T,, T ae the tems of a sequece the some times the tems T T, T T, costitute a AP/GP th tem of the seies is detemied & the sum to tems of the sequece ca easily be obtaied Remembe that to fid the sum of tems of a seies each tem of which is composed of factos i AP, the fist factos of seveal tems beig i the same AP, we wite dow the th tem, affix the ext facto at the ed, divide by the umbe of factos thus iceased ad by the commo diffeece ad add a costat Detemie the value of the costat by applyig the iitial coditios EXERCISE Q If the 0th tem of a HP is & st tem of the same HP is 0, the fid the 0 th tem ( ) Q Show that l ( 6 08 up to tems) l l Q Thee ae AM s betwee & such that 7th mea : ( ) th mea : 9, the fid the value of Q Fid the sum of the seies, 7 77 777 to tems Q Expess the ecuig decimal 076 as a atioal umbe usig cocept of ifiite geometic seies Q6 Fid the sum of the tems of the sequece a Q7 The fist tem of a aithmetic pogessio is ad the sum of the fist ie tems equal to 69 The fist ad the ith tem of a geometic pogessio coicide with the fist ad the ith tem of the aithmetic pogessio Fid the seveth tem of the geometic pogessio Q8 If the p th, q th & th tems of a AP ae i GP Show that the commo atio of the GP is q p q Q9 If oe AM a & two GM s p & q be iseted betwee ay two give umbes the show that p q apq Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom Q0 The sum of tems of two aithmetic seies ae i the atio of (7 ) : ( 7) Fid the atio of thei th tem Q If S be the sum, P the poduct & R the sum of the ecipocals of a GP, fid the value of R P S Q The fist ad last tems of a AP ae a ad b Thee ae altogethe ( ) tems A ew seies is fomed by multiplyig each of the fist tems by the ext tem Show that the sum of the ew seies is ( )(a b ) ( )ab 6 Q I a AP of which a is the Ist tem, if the sum of the Ist p tems is equal to zeo, show that the sum of the ext q tems is a (p q) q/(p ) Q(a) The iteio agles of a polygo ae i AP The smallest agle is 0 & the commo diffeece is Fid the umbe of sides of the polygo (b) The iteio agles of a covex polygo fom a aithmetic pogessio with a commo diffeece of Detemie the umbe of sides of the polygo if its lagest iteio agle is 7 Q A AP & a HP have the same fist tem, the same last tem & the same umbe of tems ; pove that the poduct of the th tem fom the begiig i oe seies & the th tem fom the ed i the othe is idepedet of Q6 Fid thee umbes a, b, c betwee & 8 such that ; (i) thei sum is (ii) the umbes, a, b ae cosecutive tems of a AP & (iii) the umbes b, c, 8 ae cosecutive tems of a GP Q7 Give that a x b y c z d u & a, b, c, d ae i GP, show that x, y, z, u ae i HP Q8 I a set of fou umbes, the fist thee ae i GP & the last thee ae i AP, with commo diffeece 6 If the fist umbe is the same as the fouth, fid the fou umbes Q9 Fid the sum of the fist tems of the sequece : Q0 Fid the th tem ad the sum to tems of the sequece : (i) 9 6 (ii) 6 Q The AM of two umbes exceeds thei GM by & HM by 7 Fid the umbes Q The hamoic mea of two umbes is The aithmetic mea A & the geometic mea G satisfy the elatio A G² 7 Fid the two umbes Q Sum the followig seies to tems ad to ifiity : (i) (iii) 7 7 0 7 0 (ii) ( ) ( ) ( ) (iv) 6 68 Q Fid the value of the sum (a) (b) i i j k j Q Fo o 0 < < /, if : x 0 cos, y (i) xyz xy z 0 si, z s 0 s s whee s is zeo if s & s is oe if s cos si the : Pove that (ii) xyz x y z EXERCISE Q The seies of atual umbes is divided ito goups (), (,, ), (, 6, 7, 8, 9), & so o Show that the sum of the umbes i the th goup is ( ) Q The sum of the squaes of thee distict eal umbes, which ae i GP is S² If thei sum is S, show that ² (/, ) (, ) Q If thee be m AP s begiig with uity whose commo diffeece is,, m Show that the sum of thei th tems is (m/) (m m ) Q If S epesets the sum to tems of a GP whose fist tem & commo atio ae a & espectively, the a ( ) pove that S S S S - a ( ) ( ) Q A geometical & hamoic pogessio have the same p th, q th & th tems a, b, c espectively Show that a (b c) log a b (c a) log b c (a b) log c 0 Q6 A compute solved seveal poblems i successio The time it took the compute to solve each successive poblem was the same umbe of times smalle tha the time it took to solve the pecedig poblem How may poblems wee suggested to the compute if it spet 6 mi to solve all the poblems except fo the fist, 7 mi to solve all the poblems except fo the last oe, ad mi to solve all the poblems except fo the fist two? Q7 If the sum of m tems of a AP is equal to the sum of eithe the ext tems o the ext p tems of the same AP pove that (m ) [(/m) (/p)] (m p) [(/m) (/)] ( p) Q8 If the oots of 0x cx x 7 0 ae i hamoic pogessio, the fid c & all the oots Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom Q9(a) Let a, a, a a be a AP Pove that : a a a a a a a a a a a a a a (b) Show that i ay aithmetic pogessio a, a, a a ² a ² a ² a ² a² K a² K [K/( K )] (a ² a² K ) Q0 Let a, a,, a, a, be a AP Let S a a a a S a a a S a a a Pove that the sequece S, S, S, is a aithmetic pogessio whose commo diffeece is times the commo diffeece of the give pogessio Q If a, b, c ae i HP, b, c, d ae i GP & c, d, e ae i AP, Show that e ab²/(a b)² Q If a, b, c, d, e be umbes such that a, b, c ae i AP ; b, c, d ae i GP & c, d, e ae i HP the: (i) Pove that a, c, e ae i GP (ii) Pove that e ( b a)²/a (iii) Q If a & e 8, fid all possible values of b The sequece a, c, d, a, a, a 98 satisfies the elatio a a fo,,, 97 ad has the sum equal to 99 Evaluate a k 9 k Q If is a oot of the equatio x² ( ac) x (a² c²) ( ac) 0 & if HM s ae iseted betwee a & c, show that the diffeece betwee the fist & the last mea is equal to ac(a c) Q (a) The value of x y z is if a, x, y, z, b ae i AP while the value of ; (/x)(/y)(/z) is / if a, x, y, z, b ae i HP Fid a & b (b) The values of xyz is / o 8/ accodig as the seies a, x, y, z, b is a AP o HP Fid the values of a & b assumig them to be positive itege Q6 A AP, a GP & a HP have a & b fo thei fist two tems Show that thei ( ) th tems will be i GP if b a bab a 7 79 Q7 Pove that the sum of the ifiite seies Q8 If thee ae quatities i GP with commo atio & S m deotes the sum of the fist m tems, show that the sum of the poducts of these m tems take two & two togethe is [/( )] [S m ] [S m ] Q9 Fid the coditio that the oots of the equatio x px qx 0 may be i AP ad hece solve the equatio x x 9x 8 0 Q0 If ax bx c 0 & a x b x c 0 have a commo oot & a/a, b/b, c/c ae i AP, show that a, b & c ae i GP Q If a, b, c be i GP & log c a, log b c, log a b be i AP, the show that the commo diffeece of the AP must be / Q If a & fo >, a a -, the show that < a a 7 < Q Sum to tems : (i) (ii) x x x ( x ) ( x ) ( x ) ( x ) ( x ) a a a a a a a a a Q I a GP the atio of the sum of the fist eleve tems to the sum of the last eleve tems is /8 ad the atio of the sum of all the tems without the fist ie to the sum of all the tems without the last ie is Fid the umbe of tems i the GP Q Give a thee digit umbe whose digits ae thee successive tems of a GP If we subtact 79 fom it, we get a umbe witte by the same digits i the evese ode Now if we subtact fou fom the huded's digit of the iitial umbe ad leave the othe digits uchaged, we get a umbe whose digits ae successive tems of a AP Fid the umbe EXERCISE Q Fo ay odd itege, ( ) ( ) l [ JEE 96, ] Q x a 6a² 0a a < y b 0b² 0b b <, fid S ab (ab)² i tems of x & y Q The eal umbes x, x, x satisfyig the equatio x x² x 0 ae i AP Fid the itevals i which ad lie [JEE 96, ] Q Let p & q be oots of the equatio x x A 0, ad let & s be the oots of the equatio x 8x B 0 If p < q < < s ae i aithmatic pogessio, the A, ad B Q a, b, c ae the fist thee tems of a geometic seies If the hamoic mea of a & b is ad that of b Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom & c is 6, fid the fist five tems of the seies [ REE '98, 6 ] Q6 Select the coect alteative(s) [ JEE '98, 8 ] (a) Let T be the th tem of a AP, fo,,, If fo some positive iteges m, we have T m & T m, the T m equals : (A) (B) (C) (D) 0 m m (b) If x, y >, z > ae i GP, the x, y, ae i : z (A) AP (B) HP (C) GP (D) oe of the above (c) Pove that a tiagle ABC is equilateal if & oly if ta A ta B ta C Q7(a) The hamoic mea of the oots of the equatio x x 8 0 is (A) (B) (C) 6 (D) 8 (b) Let a, a,, a 0, be i AP & h, h,, h 0 be i HP If a h & a 0 h 0 the a h 7 is: (A) (B) (C) (D) 6 Q8 The sum of a ifiite geometic seies is 6 ad the sum of its fist tems is 60 If the ivese of its commo atio is a itege, fid all possible values of the commo atio, ad the fist tems of the seies Q9(a) Coside a ifiite geometic seies with fist tem 'a' ad commo atio If the sum is ad the secod tem is /, the : (A) a 7, 7 (B) a, 8 (C) a, (D) a, (b) If a, b, c, d ae positive eal umbes such that a b c d, the M (a b) (c d) satisfies the elatio : (A) 0 M (B) M (C) M (D) M (c) The fouth powe of the commo diffeece of a aithmetic pogessio with itege eties added to the poduct of ay fou cosecutive tems of it Pove that the esultig sum is the squae of a itege [ JEE 000, Mais, out of 00 ] Q0 Give that, ae oots of the equatio, A x x 0 ad, the oots of the equatio, B x 6 x 0, fid values of A ad B, such that,, & ae i HP [ REE 000, out of 00 ] Q The sum of oots of the equatio ax bx c 0 is equal to the sum of squaes of thei ecipocals Fid whethe bc, ca ad ab i AP, GP o HP? [ REE 00, out of 00 ] Q Solve the followig equatios fo x ad y log x log x log 6 x y 9 ( y ) ( y ) log x [ REE 00, out of 00 ] Q(a) Let be the oots of x x p 0 ad be the oots of x x q 0 If ae i GP, the the itegal values of p ad q espectively, ae (A), (B), (C) 6, (D) 6, (b) If the sum of the fist tems of the AP,, 8, is equal to the sum of the fist tems of the AP 7, 9, 6,, the equals (A) 0 (B) (C) (D) (c) Let the positive umbes a, b, c, d be i AP The abc, abd, acd, bcd ae (A) NOT i AP/GP/HP (B) i AP (C) i GP (D) HP [JEE 00, Sceeig, out of ] (d) Let a, a be positive eal umbes i GP Fo each, let A, G, H, be espectively, the aithmetic mea, geometic mea ad hamoic mea of a, a, a, a Fid a expessio fo the GM of G, G, G i tems of A, A A, H, H, H Q(a) Suppose a, b, c ae i AP ad a, b, c ae i GP If a < b < c ad a b c, the the value of a is (A) (B) (C) (D) (b) Let a, b be positive eal umbes If a, A, A, b ae i AP ; a, a, a, b ae i GP ad a, H, H, b ae i HP, show that G G A A ( a b) ( a b) H H H H 9ab [ JEE 00, Mais, out of 60 ] Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 6 of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom c Q If a, b, c ae i AP, a, b, c ae i HP, the pove that eithe a b c o a, b, fom a GP Q6 The fist tem of a ifiite geometic pogessio is x ad its sum is The (A) 0 x 0 (B) 0 < x < 0 (C) 0 < x < 0 (D) x > 0 Q7 If a, b, c ae positive eal umbes, the pove that [( a) ( b) ( c)] 7 > 7 7 a b c Q8(a) I the quadatic equatio ax bx c 0, if b ac ad,, ae i GP whee, ae the oots of ax bx c 0, the (A) 0 (B) b 0 (C) c 0 (D) 0 (b) If total umbe of us scoed i matches is ( ) whee >, ad the us scoed i the k th match ae give by k k, whee k Fid [JEE 00 (Mais), ] Q9 If A ad B A, the fid the miimum atual umbe 0 such that B > A > 0 [JEE 006, 6] EXERCISE Pat : (A) Oly oe coect optio If x R, the umbes x x, a/, x x fom a AP the 'a' must lie i the iteval: (A) [, ] (B) [, ] (C) [, ] (D) [, ) a b c If x > ad,, x x x ae i GP, the a, b, c ae i (A) AP (B) GP (C) HP (D) oe of these If A, G & H ae espectively the AM, GM & HM of thee positive umbes a, b, & c, the the equatio whose oots ae a, b, & c is give by: (A) x Ax G x G 0 (B) x Ax (G /H)x G 0 (C) x Ax (G /H) x G 0 (D) x Ax (G /H) x G 0 The sum is equal to: (A) (B) / (C) / (D) oe If a, a, a, a,, a, b ae i AP ad a, g, g, g,g, b ae i GP ad h is the hamoic mea of a a a a a a a ad b, the g g g g g g is equal to (A) h (B) h (C) h (D) h 6 Oe side of a equilateal tiagle is cm The midpoits of its sides ae joied to fom aothe tiagle whose mid poits ae i tu joied to fom still aothe tiagle This pocess cotiues idefiitely The the sum of the peimetes of all the tiagles is (A) cm (B) cm (C) 88 cm (D) oe of these 7 If p is positive, the the sum to ifiity of the seies, p ( p) - is: p ( p) ( p) (A) / (B) / (C) (D) oe of these 8 I a GP of positive tems, ay tem is equal to the sum of the ext two tems The commo atio of the GP is (A) cos 8 (B) si 8 (C) cos 8 (D) si 8 9 If upto 6, the (A) / (B) / (C) /8 (D) oe of these 0 The sum to 0 tems of the seies 6 8 is (A) ( 6 ) (B) ( ) (C) ( ) (D) ( ) If a, a, a ae i AP with commo diffeece d 0, the the sum of the seies (si d) [cosec a cosec a cosec a cosec a cosec a cosec a ] (A) sec a sec a (B) cosec a cosec a (C) cot a cot a (D) ta a ta a Sum of the seies S 00 00 is (A) 007006 (B) 0000 (C) 00006 (D) oe of these If H, the value of is (A) H (B) H (C) H (D) H The sum of the seies log log log8 log is (A) ( ) (B) ( ) ( ) (C) (D) ( ) ( ) Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 7 of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom If S, S, S ae the sums of fist atual umbes, thei squaes, thei cubes espectively, the S( 8S) is equal to S (A) (B) (C) 9 (D) 0 6 If p ad q ae espectively the sum ad the sum of the squaes of successive iteges begiig with a, the q p is (A) idepedet of a (B) idepedet of (C) depedet o a (D) oe of these 7 x x(x a) x(x a)(x a ) Sum of tems of the seies a aa aa a is x(x a) (x a) (x a)(x a ) (x a) x(x a) (x a ) (A) (B) a a a a a a (C) a a a (D) oe of these 8 {a } ad {b } ae two sequeces give by a ( x) ( y) ad b ( x) ( y) fo all N The value of a a a a is equal to x y x y xy (A) x y (B) (C) (D) b b b 9 If a, a, a,, a ae positive eal umbes whose poduct is a fixed umbe c, the the miimum value of a a a a a is [IIT - 00, ] (A) (c) / (B) ( ) c / (C) c / (D) ( )(c) / Pat : (B) May have moe tha oe optios coect 0 If ( ) ( ) a b c d e, the (A) a c b d (B) e 0 (C) a, b /, c ae i AP (D) c/a is a itege The sides of a ight tiagle fom a GP The taget of the smallest agle is (A) (B) (C) (D) Sum to tems of the seies S () ( ) (6 ) is (A) ( ) whe is eve (C) ( ) whe is odd If a, b, c ae i HP, the: a b c (A),, b c a c a b a b c (C) a b, b, c b (B) ( ) whe is odd (D) ( ) whe is eve ae i HP (B) b b a ae i GP (D) a b c b c, b c, c a a b ae i HP If b, b, b (b i > 0) ae thee successive tems of a GP with commo atio, the value of fo which the iequality b > b b holds is give by (A) > (B) < (C) (D) EXERCISE If a, b, c ae i AP, the show that: (i) a (b c), b (c a), c (a b) ae also i AP(ii) b c a, c a b, a b c ae i AP If a, b, c, d ae i GP, pove that : (i) (a b ), (b c ), (c d ) ae i GP (ii),, a b b c c d ae i GP Usig the elatio AM GM pove that (i) ta cot ; if 0 < < (ii) (x y y z z x) (xy yz zx ) > 9x y z (iii) (a b) (b c) (c a) abc ; if a, b, c ae positive eal umbes Fid the sum i the th goup of sequece, (i), (, ); (,, 6, 7); (8, 9,, ); (ii) (), (,, ), (, 6, 7, 8, 9), If is a oot of the equatio x² ( ac) x (a² c²) ( ac) 0 & if HM s ae iseted betwee a & c, show that the diffeece betwee the fist & the last mea is equal to ac(a c) 6 The sum of the fist te tems of a AP is & the sum of fist two tems of a GP is 9 The fist tem of the AP is equal to the commo atio of the GP & the fist tem of the GP is equal to the commo diffeece of the AP Fid the two pogessios 7 Fid the sum of the seies () () () up to 8 If 0 < x < ad the expessio exp {( cos x cos x cos x cos x upto ) log e } satisfies the quadatic equatio y 0y 6 0 the fid the value of x 9 I a cicle of adius R a squae is iscibed, the a cicle is iscibed i the squae, a ew squae i the cicle ad so o fo times Fid the limit of the sum of aeas of all the cicles ad the limit of the sum of aeas of all the squaes as 0 The sum of the squaes of thee distict eal umbes, which ae i GP is S² If thei sum is S, show that ² (/, ) (, ) Let S, S,S p deote the sum of a ifiite GP with the fist tems,,, p ad commo atios Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 8 of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't
Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom /, /,, /(p ) espectively Show that S S S p p(p ) Cicles ae iscibed i the acute agle so that evey eighbouig cicles touch each othe If the adius of the fist cicle is R the fid the sum of the adii of the fist cicles i tems of R ad Give that ae oots of the equatio, A x x 0 ad, the oots of the equatio, B x 6 x 0, fid values of A ad B, such that,, & ae i HP The aithmetic mea betwee m ad ad the geometic mea betwee a ad b ae each equal to ma b : fid the m ad i tems of a ad b m If a, b, c ae positive eal umbes the pove that (i) b c c a a b > abc (a b c) (ii) (a b c) > 7abc (iii) (a b c) > 7 (a b c) (c a b) (b c a) s s s 6 If 's' be the sum of '' positive uequal quatities a, b, c,, the s a s b s c 7 Sum the followig seies to tems ad to ifiity: (i) (ii) (iii) Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't 6 ( ) ( ) ( ) 7 7 9 8 Let a, b, c d be eal umbes i GP If u, v, w, satisfy the system of equatios u v w 6; u v 6w 6u 9v the show that the oots of the equatio 7 7 9 x u v w [(b c) (c a) (d b) ] x u v w 0 ad 0x 0 (a d) x 9 0 ae ecipocals of each othe [IIT- 999, 0] 9 The fouth powe of the commo diffeece of a aithmetic pogessio with itege eties added to the poduct of ay fou cosecutive tems of it Pove that the esultig sum is the squae of a itege [IIT - 000, ] 0 If a, b & c ae i aithmetic pogessio ad a, b & c ae i hamoic pogessio, the pove that eithe a b c o a, b & c ae i geometic pogessio [IIT 00, ] ANSWER KEY EXERCISE Q Q µ Q S (7/8){0 9 0} Q / Q 6 ( )/ (² ) Q 7 7 Q 0 ( 6)/(8 ) Q Q (a) 9 ; (b) Q 6 a, b 8, c Q 8 (8,,, 8) Q 9 ² Q 0 (i) ; (ii) ² ; (/6) ( ) ( ) Q 0, 0 Q 6, Q (i) s (/) [/{6( ) ( ) }] ; s / (ii) (/) ( ) ( ) ( ) ( ) ( )( ) (iii) /( ) (iv) S ; S 6()( ) Q (a) (6/) (6 ) (b) [( ) ( )]/6 EXERCISE Q 6 8 poblems, 7 miutes Q8 C 9 ; (, /, /) Q (iii) b, c 6, d 9 OR b, c 6, d 8 Q 99 Q (a) a, b 9 OR b, a 9 ; (b) a ; b o vice vesa Q9 p 9pq 7 0; oots ae,, 7 Q (a) x ( x ) ( x ) ( x ) (b) ( a ) ( a ) ( a ) Q 8 Q 9 EXERCISE Q ( ) ( )² Q S ab ( ab) Whee a x / & b y / Q (/) ; (/7) Q, 77 Q 8,, 7, 6, 68 Q 6 (a) C (b) B Q 7 (a) B (b) D Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 9 of 6
FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom Q 8 ± /9 ; ; a /80 OR ± / ; ; a 08 OR /8 ; ; a 60 Q 9 (a) D (b) A Q 0 A ; B 8 Q AP Q x ad y Q (a) A, (b) C, (c) D, (d) A, A, A H, H, H Q (a) D Q6 B Q8 (a) C, (b) 7 Q9 0 EXERCISE D A B B A 6 A 7 A 8 D 9 C 0 A C A A D C 6 A 7 B 8 C 9 A 0 ABCD BC AB ABCD ABCD EXERCISE (i) ( ) (ii) ( ) 6 ( 6 ); (/ / 6/6 ) 7 9 R ; R R si si b a a b m a b, a b si si 7 (i) (/) ( ) ( ) ( ) ( ) (ii) (iii) ( ) 9 ( ) ( ) 6 6 8 A ; B 8 ( ) ; s ( ),, Teko Classes, Maths : Suhag R Kaiya (S R K Si), Bhopal Phoe : 0 90 90 7779, 0 9890 888 page 0 of 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will" Ieffective People do't