Chapte 7 Sequece ad seies Chapte Sequeces ad Seies. Itoductio: The INVENTOR of chess asked the Kig of the Kigdom that he may be ewaded i lieu of his INVENTION with oe gai of wheat fo the fist squae of the boad, two gais fo the secod, fou gais fo the thid, eight gais fo the fouth, ad so o fo the sixty fou squaes. Fotuately, this appaetly modest equest was examied befoe it was gated. By the twetieth squae, the ewad would have amouted to moe tha a millio gais of wheat; by the sixty-fouth squae the umbe called fo would have bee astoomical ad the bulk would have fo exceeded all the gais i the kigdom. The basis of this stoy a sequece of umbes that have a mathematical elatioship --- has a geat may impotat applicatios. May of them ae beyod the scope of this book, but we shall exploe the meas of dealig with a umbe of pactical, ad ofte etetaiig, poblems of this type.. Sequeces: A set of umbes aaged i ode by some fixed ule is called as sequeces. Fo example (i), 4, 6, 8, 0,, 4 - - - - - - - - - - - - (ii), 3, 5, 7, 9, - - - - - - - - - - - - (iii), 4, 8, 6, - - - - - - - - - - - - I sequece a, a, a 3, a,.. a is the fist tem, a is the secod tem, a 3 is the thid ad so o. A sequece is called fiite sequece if it has fiite tems e.g.,, 4, 6, 8, 0,, 4, 6. A sequece is called ifiite sequece if it has ifiite tems, e.g.,, 4, 6, 8, 0,, 4, - - - - - - - -.3 Pogessio: If a sequece of umbe is such that each tem ca be obtaied fom the pecedig oe by the opeatio of some law, the sequece is called a pogessio. Note:- Each pogessio is a sequece but each sequece may o may ot be a pogessio.4 Aithmetic Sequece:
Chapte 8 Sequece ad seies A sequece i which each tem afte the fist tem is obtaied fom the pecedig tem by addig a fixed umbe, is called as a aithmetic sequece o Aithmetic Pogessio, it is deoted by A.P. e.g., (i), 4, 6, 8, 0,, - - - - - - - - - (ii), 3, 5, 7, 9,, - - - - - - - - - - Commo Diffeece: The fixed umbe i above defiitio is called as commo diffeece. It is deoted by d. it is obtaied by subtactig the pecedig tems fom the ext tem i.e ; a a - ;. Fo example, 4, 6, 8, 4, - - - - - - - - d = Commo diffeece = a a = 4 = O d = Commo diffeece = a 3 a = 6 4 = The Geeal Fom of a Aithmetic Pogessio: Let a be the fist tem ad d be the commo diffeece, the Geeal fom of a aithmetic pogessio is a, a + d, a + d, - - - - - - - - - - a + ( )d.5 th tem o Geeal tem(o, last tem)of a Aithmetic Pogessio: If a be the fist tem ad d be the commo diffeece the a = fist tem = a = a + ( )d a = d tem = a + d = a + ( )d a 3 = 3d tem = a + d = a + (3 )d a 4 = 4th tem = a + 3d = a + (4 )d - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - a = th tem = a + ( )d a = th tem = a + ( )d i which a = st tem d = commo diffeece = umbe of tems Example : Fid the 7th tem of A.P. i which the fist tem is 7 ad the commo diffeece is 3. Solutio: a 7 = 7th tem =? a = 7 d = 3 Puttig these values i a = a + ( )d a 7 = 7 + (7 )( 3)
Chapte 9 Sequece ad seies = 7 + 6 ( 3) = 7 8 a 7 = Example : 5 3 Fid the 9th tem of the A.P.,,,...... 4 4 4 Solutio: 5 a = 4 5 5 d = 4 4 4 4 5 4 = 4 4 9th tem = a 9 =? a = a + ( )d 5 a 9 = (9 ). 4 5 = 8 4 5 3 7 a 9 = 4 4 Example 3: ( ) Fid the sequece whose geeal tem is Solutio: ( ) Hee a = Put =, a = ( ) 0 0 Put =, a = ( ) () Put = 3, a 3 = 3(3 ) 3() 3 Put = 4, a 4 = 4(4 ) 4(3) 6
Chapte 30 Sequece ad seies Put = 5, a 5 = 5(5 ) 5(4) 0 Theefoe the equied sequece is 0,, 3, 6, 0, - - - - - - - - Execise. Q. Fid the tems idicated i each of the followig A.P. (i), 4, 7, - - - - - - - - 7th tem (ii) 7, 7, 7, - - - - - - - - 3th tem (iii),,, - - - - - - - - 6 4 3 0th tem Q. Fid the fist fou tems of A.P. i which fist tem is 7 ad commo diffeece is 4. Q.3 Fid the umbe of tems i a A.P. i which a = 5, d = 5 ad a = 30. Q.4 (i) Which tem i the aithmetic pogessio 4,, -. is 77? (ii) Which tem i the aithmetic pogessio 7, 3, 9. is 9? Q.5 Fid the 7th tem of a A.P. whose 4th tem is 5 ad the commo diffeece is. Q.6 What is the fist tem of the eight tem A.P. i which the commo diffeece is 6 ad the 8th tem is 7. Q.7 Fid the 0th tem of the A.P. whose 3d tem is 7 ad 8th tem is 7. Q.8 If the th tem of a A.P. is 9 ad 7th tem i 9, Fid the fist tem ad the commo diffeece. Q.9 The 9th tem of a A.P. is 30 ad the 7th tem is 50. Fid the fist thee tems. Q.0 Fid the sequece whose th tem is 4 + 5. Also pove that the sequece is i A.P. Q. If ae i A.P, show that b = Q. If ae i A.P, show that the commo diffeece is Aswes. Q. (i) a 7 = 9 (ii) a 3 = 7 (iii) a 0 = 7 4 Q. 7, 3,, 5 Q.3 = 6 Q.4 (i) = 8 (ii) = 0 Q.5 a 7 =
Chapte 3 Sequece ad seies Q.6 a = 5 Q.7 a 0 = 4 Q.8 a = 3, d = Q.9 0, 5, 5 Q.0 9, 3, 7, - - - - - - ad the diffeece betwee cosecutive tems is equal. So the sequece is a A.P..6 Aithmetic Meas (A.Ms): If a, A, b ae thee cosecutive tems i a Aithmetic Pogessio, The A is called the Aithmetic Mea (A.M) of a ad b. i.e. if a, A, b ae i A.P. the A a = b A A + A = a + b A = a + b A = a + b The aithmetic mea of two umbes is equal to oe half the sum of the two umbes. Example : Fid the A.M. betwee 5 4 ad 5 4 Solutio: Hee a = 5 4, b = 5 4 A.M. = A = a + b A = 5 4 + 5 4 = 5 = 5.7 Aithmetic Meas betwee a ad b: The umbe A, A, A 3 - - - - - A ae said to be aithmetic meas betwee a ad b if a, A, A, A 3, - - - - - - A, b ae i A.P. We may obtai the aithmetic meas betwee two umbes by usig a = a +( )d to fid d, ad the meas ca the be computed. Example : Iset thee A.M s betwee 8 ad 4. Solutio: Let A, A, A 3 be the equied A.M s betwee 8 ad 4, the 8, A, A, A 3, 4 ae i A.P. Hee a = 8, = 5, a 5 = 4, d =? Usig a = a + ( )d
Chapte 3 Sequece ad seies a 5 = 8 + (5 )d 4 = 8 + 4d 4d = 4 + 8 4d = d = Theefoe A = d tem = a + d = 8 + = 5 5 A = 3d tem = A + d = + 5 = 4 A = = 7 5 3 Thus the equied A.M s ae, 7, Example 3: Iset A.M s betwee a ad b. Solutio: Let, A, A, A 3, A be the. A.M s betwee a ad b. The a, A, A, A 3, b, ae i A.P. Let, d be the commo diffeece So, a = a, = +, d =? a = b a = a + ( )d b = a + ( + )d b = a + ( + )d b a = ( + )d d = b a + A = a + d = a + b a a( + )+b a = + + A = a + b + A = A + d = a + b a + + b a + a + b a = + + a + a + b a = +
Chapte 33 Sequece ad seies A = ( )a + b + [-( )]a + b ( +)a + b a + b A =.... + + + Thus A.M s betwee a ad b ae: a + b ( )a + b ( )a + 3b a + b,,.... + + + + Execise. Q. Fid the A.M. betwee (i) 7 ad 3 (ii) 5 ad 40 (iii) + 3 ad 3 (iv) x + b ad x b Q. Iset two A.M s betwee 5 ad 40. Q.3 Iset fou A.M s betwee ad 3 Q.4 Iset five A.M s betwee 0 ad 5. Q.5 Iset six A.M s betwee ad 9. Q.6 If 5, 8 ae two A.M s betwee a ad b, fid a ad b + + Q.7 a + b Fid the value of so that a + b may be the A.M s betwee a ad b. Q.8 Fid the value of x if x +, 4x + ad 8x ae the cosecutive tems of a aithmetic pogessio. Q.9 Show that the sum of A.M s betwee a ad b is equal to times thei sigle A.M. Q. (i) 7 (ii) Q. 0, 5 Q.3 Q.4 5 35 45,5,,0, Aswe. 35 7 9 3,,, 0 0 0 0 (iii) (iv) x Q.5 9, 6, 3, 0 3, 6 Q.6 a=, b = Q.7 = 0 Q.8 x =
Chapte 34 Sequece ad seies Seies: The sum of the tems of a sequece is called as seies. Fo example:, 4, 9, 6, - - - - - - is a sequece. Sum of the tems of sequece i.e., + 4 + 9 + 6 - - - - - epeset a seies..8 Aithmetic Seies: The sum of the tems of a Aithmetic sequece is called as Aithmetic seies. Fo example: 7, 7, 7, 37, 47, - - - - - - - - - is a A.P. 7 + 7 + 7 + 37 + 47 + - - - - - - - - is Aithmetic seies. The sum of tems of a Aithmetic Sequece: The geeal fom of a aithmetic sequece is a, a + d, a + d, - - - - - - - - a + ( )d. Let S deoted the sum of tems of a Aithmetic sequece. The S = a + (a + d)+(a + d) + - - - - + [a +( )d] Let th tem [a + ( )d] = The above seies ca be witte as S = a+(a + d) + (a + d) + - - - - - + O, S = a + (a + d) + (a + d) + - - - - - + ( - d) + ( - d) + -----(I) Witig i evese ode, we have S = + ( - d) + ( - d) + - - -(a + d) + (a + d) + a (II) Addig I ad II S = (a + ) + (a + ) + (a + ) + - - - - + (a + ) S = (a + ) S = (a + ) But = a + ( )d S = [a + (a + ( )d)] = [a + a + ( )d] S = [a + ( )d] is the fomula fo the sum of tems of a aithmetic sequece. Example : Fid the sum of the seies 3 + + 9 + - - - - to 6 tems. Solutio: Hee a = 3, d = 3 = 8, = 6
Chapte 35 Sequece ad seies Usig fomula S = [a + ( )d] S 6 = 6 [(3) + (6 )8] = 8[6 + 5(8)] = 8[6 + 0] S 6 = 8 x 6 = 008 Example : Fid the sum of all atual umbes fom to 500 which ae divisible by 3. Solutio: The sequece of umbes divisible by 3 is 3, 6, 9,, - - - - - 498 (which is i A.P.) Hee a = 3, d = 6 3 = 3, =? a = 498 Fist we fid Fo this usig a = a + ( )d 498 = 3 + ( )(3) 498 = 3 + 3 3 3 = 498 = 66 Now a = 3, d = 3, = 66, S =? S = [a + ( )d] S 66 = 66 [(3) + (66 )3] = 83[6 + 65(3)] = 83(6 + 495) = 83 x 50 S 66 = 4583 Example 3: ` If the sum of tems of a A.P. is + 3. Fid the th tem. Solutio: We have S = + 3 S - = ( ) +3( ) = ( ) +3( + ) = + 3 6 + 3 S - = 3 4 + th tem = a = S S - = + 3 (3 4 + ) = + 3 3 + 4 a = 6
Chapte 36 Sequece ad seies Example 4: The sum of thee umbes i a A.P. is ad the sum of thei cubes is 408. Fid them. Solutio: Let the equied umbes be a d, a, a + d Accodig to st coditio: (a d) + a + (a + d) = a d + a + a + d = 3a = a = 4 Accodig to d give coditio: (a d) 3 + a 3 + (a + d) 3 = 408 a 3 d 3 3a d + 3ad + a 3 + a 3 + d 3 + d 3 + 3a d + 3ad = 408 3a 3 + 6ad = 408 3(4) 3 + 6(4) d = 408 4d = 408 9 d = 9 d = 3 Whe a = 4 d = 3 the umbe ae a d, a, a + d i.e. 4 3, 4, 4 + 3 i.e., 4, 7 whe a = 4, d = 3 the umbes ae a d, a, a + d 4 ( 3), 4, 4 + ( 3) 4 + 3, 4, 4 3 i.e., 7, 4, Hece the equied umbes ae, 4, 7 o 7, 4, Note: The poblem cotaiig thee o moe umbes i A.P. whose sum i give it is ofte to assume the umbe as follows. If the equied umbes i A.P ae odd i.e. 3, 5, 7 etc. The take a (fist tem) as the middle umbe ad d as the commo diffeece. Thus thee umbes ae a d, a, a + d. If the equied umbes i A.P ae eve i.e., 4, 6, etc. the take a d, a + d as the middle umbes ad d as the commo diffeece. Thus fou umbes ae a 3d, a d, a + d, a + 3d ad six umbes ae: a 5d, a 3d, a d, a + d, a + 3d, a + 5d etc. Example 5: A ma buys a used ca fo $600 ad agees to pay $00 dow ad $00 pe moth plus iteest at 6 pecet o the outstadig idebtedess util the ca paid fo. How much will the ca cost him? Solutio:
Chapte 37 Sequece ad seies The ate of 6 pecet pe yea is 0.5 pecet pe moth. Hece, whe the puchase makes his fist paymet, he will owe moth s iteest. The iteest o $500 = (500)(0.005) = $.50 The puchase will pay i the secod moth = $0.50 Sice the puchase pays $00 o the picipal, his iteest fom moth to moth is educed by 0.5 pecet of $00, which is $0.50 pe moth. The fial paymet will be $00 plus iteest o 00 fo moth, which is = $00.50 Hece his paymets o $500 costitute a aithmetic pogessio 0.50 + - - - - - - + 00.50 Hee a = 0.40, = 00.50 ad = 5 Theefoe by the fomula S = (a + ) = 5 (0.50 + 00.50) = 5 (03) = $507.50 Thus, the total cost of the ca will be $ 607.50 Execise.3 Q. Sum the seies: (i) 5 + 8 + + 4 +.. to tems. (ii) 5 + 50 + 49 + +. (iii) 5 + 3 + to 0 tems. (iv) +-------- to tems Q. The th tem of a seies i 4 +. Fid the sum of its st tems ad also the sum of its fist huded tems. Q.3 Fid the sum of the fist 00 odd positive iteges. Q.4 Fid the sum of all the itegal multiples of 3 betwee 4 ad 97 Q.5 How may tems of the seies: (i) - 9-6 - 3 amout to 66? (ii) 5 + 7 + 9 + amout to 9? Q.6 Obtai the sum of all the iteges i the fist 000 positive iteges which ae eithe divisible by 5 o. Q.7 The sum of tems of a seies is 7 + 8. Show that it is a A.P ad fid its commo diffeece.
Chapte 38 Sequece ad seies Q.8 Sum the seies + 3 5 + 7 + 9 + 3 + 5 7 to 3 tems. Q.9 If S, S, S 3 be sums to,, 3 tems of a aithmetic pogessio, Show that S 3 = 3(S S ). Q.0 The sum of thee umbes i A.P. is 4, ad thei poduct is 440. Fid the umbes. Q. Fid fou umbes i A.P. whose sum is 4 ad the sum of whose squae is 64. Q. Fid the five umbe i A.P. whose sum is 30 ad the sum of whose squae is 90.0 Q.3 How may bicks will be thee i a pile if thee ae 7 bicks i the bottom ow, 5 i the secod ow, etc., ad oe i the top ow? Q.4 A machie costs Rs. 300, depeciates 5 pecet the fist yea, pecet of the oigial value the secod yea, 7 pecet of the oigial value of the thid yea, ad so o fo 6 yeas. What is its value at the ed of 6 yeas. Aswes.3 Q. (i)s = [3 + 7] (ii) = 3, S = 6 (iii) S 0 = 40 (iv) Q. S = ( + 3), S 00 = 0300 Q.3 S 00 = 40,000 Q.4 58 Q.5 (i) = (ii) = Q.6 000,000 Q.7 d = 4 Q.8 S 3 = (3 4) Q.0 5, 8, o, 8, 5 Q. 3, 5, 7, 9 o 9, 7, 5, 3 Q. 4, 5, 6, 7, 8 o 8, 7, 6, 5, 4 Q.3 96 Q.4 Rs. 30.00.9 Geometic Sequece o Pogessio (G.P): A geometic pogessio is a sequece of umbes each tem of which afte the fist is obtaied by multiplyig the pecedig tem by a costat umbe called the commo atio. Commo atio is deoted by. Example: (i) (ii) (iii), 4, 8, 6, 3, is G.P because each umbe is obtaied by multiplyig the pecedig umbe by., 4, 8, 4,, 36,
Chapte 39 Sequece ad seies Note:- I geometic pogessio, the atio betwee ay two cosecutive tems emais costat ad is obtaied by dividig the ext tem with the a peceedig tem, i.e., =, a -.0 th tem o Geeal tem(o, last tem) of a Geometic Pogessio (G.P): If a is the fist tem ad is the commo atio the the geeal fom of G.P is a, a, a, a 3, If a = st tem = a a = d tem = a a 3 = 3 d tem = a - - - - - - - - - - - - - - - - - - - - - - - - - - a = th tem = a - Which is the th tem of G.P i which: a = st tem = commo atio = umbe of tems a = th tem = last tem Example : Wite the sequece i which a = 5, = 3, = 5 a = a = 5 a = a = 5(3) = 5 a 3 = a = 5(3) = 45 a 4 = a 3 = 45(3) = 35 a 5 = a 4 = 35(3) = 405 Theefoe, the equied sequece is: 5, 5, 45, 35, 405 Example 3: Fid 4th tem i the G.P. 5, 0, 0, Solutio: a = 5, = 0 5 =, a =? a = a - a 4 = a 4 = 5() 4- = 5 x 8 = 40 Example 4: Fid i the G.P. 4,,, if a = 6 Solutio: Sice 4,,,
Chapte 40 Sequece ad seies Hee, a = 4, =, 4 =, a = 6 a = a - 6 = 4( )- let 6 x 4 = ( )- = 64 = ( )- ( )6 = ( )- = 6 o = 6 + = 7 Example 5: Fid the G.P. of which the thid tem is 4 ad 6th is 3. Solutio: Hee a 3 = 4, a 6 = 3 a = a - a 3 = a 3, a 6 = a 6 4 = a (i) 3 = a 5 (ii) Dividig (i) by (ii) a 4 o = 5 3 a 3 8 3 = 8 = ( ) 3 = Example 6: The populatio of a tow iceases at the ate of 0% aually. Its peset populatio is,00,000 what will be its populatio at the ed of 5 yeas? Solutio: Let, peset populatio = a =,00,000 (give) The icease of populatio at the ed of st yea = a(0/00) = a(0.) Total populatio at the ed of st yea = a + a(0.) = a(.) Total populatio at the ed of d yea = a(.)(.) = a(.) The populatio at the ed of 5yeas is the 6 th tems of G.P a, a(.), a(.) Hee a =,00,000, =., =6, a 6 =? Sice, a = a - a 6 =,00,000 (.) 5 =,00,000 (.605) = 30 Example 7:
Chapte 4 Sequece ad seies The value of a auto mobile depeciate at the ate of 5% pe yea. What will be the value of a automobile 3 yeas hece which is ow puchased fo Rs. 45,000? Solutio: a = 45,000 = Puchased value of automobile The amout depeciate at the ed of st yea = a (5/00) = 0.5a The value of automobile at the ed of st yea = a 0.5a = a( 0.5) = a(0.85) The value of automobile at the ed of d yea = a(0.85)( 0.5) = a(0.85)(0.85) = a(0.85) The value of automobile at the ed of 3d yea = a(0.85) 3 Execise.4 = 45,000(0.85) 3 = 45,000 (0.645) = 7635.63 upees Q. Wite the ext five tems of the followig G.Ps. (i), 0,. (ii) 7, 9, 3,. (iii),,, 4. Q. Fid the tem idicated i each of the followig G.Ps. (i), 3 3, 3 6,., 6th tem (ii) i,, i,,., 3th tem (iii), 6,3,., 5th tem (iv),,., 6th tem 3 9 7 Q.3 Fid the th tem of the G.P. (i) a = 8, = 3, = 5 (ii) a =, = 4, = 6 (iii) a = 3, =, = 0 Q.4 Wite dow the fiite geometic sequece which satisfies the give coditio. (i) a = 3, = 5, = 6 (ii) Fist tem =, secod tem = 6, = 5 (iii) Thid tem = 9, sixth tem = 3, = 8 (iv) Fifth tem = 9, eight tem = 43, = 8
Chapte 4 Sequece ad seies Q.5 If ae G.P,show that the commo atio is Q.6 If the secod tem of a G.P is ad the th tem is, what is 56 the fist tem? What is the th tem. Q.7 Fid the 0th tem of a G.P if d tem 43 ad 4th tem is 9. Q.8 What is the fist tem of a six tem geometic pogessio i which the atio is 3 ad the sixth tem is 7? Q.9 A busiess coce pays pofit at the ate of 5% compouded aually. If a amout of Rs.,00,000 is ivested with the coce ow, what total amout will become payable at the ed of 5 yeas? Q.0 A ubbe bell is dopped fom a height of 6dm, it cotiuously ebouds to 3 4 of the distace of its pevious fall. How high does it eboud its fouth time? Q. Fid thee cosecutive umbes i G.P whose sum is 6 ad thei poduct is 6. Q. If the sum of the fou umbes cosecutive umbes of a G.P is 80 ad A.M betwee secod ad fouth of them is 30.Fid the tems. Aswes.4. (i) 50, 50, 50, 650, 350 (ii) (iii),,,, 8 6 3 64 8. (i) (7) 5 (ii) i (iii) 3. (i) 8 (ii) 04 (iii) 536 4. (i) 3, 5, 75, 375, 875, 9375 (ii), 6, 8, 54, 6 (iii) 8, 7, 9, 3,,,, 3 9 7 (iv),,, 3, 9, 7, 8, 43 9 3 6. a = 4, a = 4-7. 43,,,, 3 9 7 8 7 (3) (iv) 79
Chapte 43 Sequece ad seies 8. a = 3 9. 407.44 0. 5.06 dm., 6, 8 o 8, 6,., 6, 8, 54. Geometic Mea: Whe thee quatities ae i G.P., the middle oe is called the Geometic Mea (G.M.) betwee the othe two. Thus G will be the G.M. betwee a ad b if a, G, b ae i G.P. To Fid G.M betwee a ad b: Let, G be the G.M. betwee a ad b The a. G. b ae i G.P G b G = ab a G G = ab Hece the G.M. betwee two quatities is equal to the squae oot of thei poduct. Example : Fid the G.M. betwee 8 ad 7. Solutio: G = ab G = 8 x 7 8 x 8 x 9 8x 3 G = 4. G.Ms Betwee a ad b: The umbes G, G, G 3 G ae said to be G.Ms betwee a ad b if a, G, G, G 3 G, b ae i G.P. I ode to obtai the G.M s betwee a ad b, we use the fomula a = a - to fid the value of ad the the G.M s ca be computed. To Iset G.M s Betwee Two Numbes a ad b Let,G, G, G 3 G be G.Ms betwee a ad b Hee a = a, a = b, = +, =? a = a - b = a -- = a - b/a = - = (b/a) /(-) So, G = a = a(b/a) /(+) G = G = a(b/a) /(+) (b/a) /(+) = a(b/a) /(+) G 3 = G = a(b/a) /(+) (b/a) /(+) = a(b/a) 3/(+) G = a(b/a) /(+) Example : Fid thee G.M s betwee ad 3. Solutio:
Chapte 44 Sequece ad seies Let,G, G, G 3 G be G.Ms betwee ad 3 The, G, G, G 3 3 ae i G.P. Hee a =, a = 3, =? = 5 a = a - 3 = () 5- = 4 6 = 4 4 = 4 = So, G = a = () = 4 G = G = 4() = 8 G 3 = G = 8() = 6 Thus thee G.M s betwee ad 3 ae 4, 8, 6. Example 3: Iset 6 G.M s betwee ad 56. Solutio: Let,G, G, G 3, G 4, G 5, G 6 be six G.M s betwee ad 56. The, G, G, G 3, G 4, G 5, G 6 56 ae i G.P. Hee a =, a = 3, =? = 5 a = a - 56 = () 8- = 7 8 = 7 () 7 = = So, G = a = () = 4 G = G = 4() = 8 G 3 = G = 8() = 6 G 4 = G 3 = 6() = 3 G 5 = G 4 = 3() = 64 G 6 = G 5 = 64() = 8 Hece, equied G.M s ae 4, 8, 6, 3, 64, 8. Example 4: The A.M betwee two umbes is 0 ad thei G.M is 8. Detemie the umbes. Solutio: A.M = a + b = 0 a + b = 0....... () G.M. = ab 8 ab = 64....... () fom () b = 64 a, Put i ()
Chapte 45 Sequece ad seies a + 64 = 0 a + 64 = 0a (a 6)(a 4) = 0 a = 6 o a = 4 Whe, a = 6, b = 64 6 = 4 Whe, a = 4, b = 64 6 = 6 Hece the umbes ae 4 ad 6. Execise.5 Q. Fid G.M betwee (i) 4, 64 (ii), 43 3 Q. Iset two G.M s betwee ad. Q3. Iset thee G.M s betwee 56 ad. Q4. Iset fou G.M s betwee 9 ad 7. (iii) 8 8, 9 9 Q5. Show that A.M of two uequal positive quatities is geate tha this G.M. + + a b Q6. Fo what value of is the G.M betwee a ad b, a b whee a ad b ae ot zeo simultaeously. Q7. Pove that the poduct of G.M s betwee a ad b is equal to the powe of the sigle G.M betwee them. Q8. The A.M of two positive itegal umbes exceeds thei (positive)g.m by ad thei sum is 0.Fid the umbes. Aswes.5 Q. (i) 6 (ii) 9 (iii) 8 9 /3 5/6 Q., Q3. 64, 6, 4 Q4. 3,,, 3 9 Q6. = Q8. 6, 4 o 4, 6
Chapte 46 Sequece ad seies.3 Geometic Seies A geometic seies is the sum of the tems of a geometic sequece. If a, a, a, + a - is a geometic sequece. The a + a + a + + a - is a geometic seies. Sum of Tems of a Geometic Seies Let, S be the sum of geometic seies i.e. S = a + a + a + + a -.. () Multiplyig by o both sides S = a + a + a 3 + + a - +a.. () Subtactig () fom (), we get S S = a a ( )S = a( ) a( ) S = ; Fo coveiece, we use : a( ) S = if a( ) ad S = if > Example : Sum the seies,, 3,......... to 7 tems 3 Solutio Hee a = 3, = 3 = 3 a( ) S = (because < ) 7 3 87 3 3 8 S 7 = 3 5 35 463 S 7 = 3 8 5 96
Chapte 47 Sequece ad seies Example : Sum to 5 tems the seies + 3 + 9 +.. Solutio: The give seies is a G.P. a 3 i which a =, = 3, = 5 a 4 a( ) S = (because > ) 5 [(3) ] 43 4 S = 3 Example 3: Fid S fo the seies + 4 + 8 +.. +. Sice = > a( ) ( ) + S = Example 4: How may tems of the seies 55........ amout to 3 3 7 Solutio: S = 55 7, =? a = 9, = 3 3 9 a( ) S = 3 3 55 9 9 7 3 3 55 4 3 7 45 45x55 3 75 3 7x4 3
Chapte 48 Sequece ad seies 5 3 75 43 3 3 3 = 5 Example 5: Sum the seies: (i) 0. +. +. + to tems (ii) (x + y)(x + xy + y ) + (x 3 + x y + xy + y 3 ) + to tems. Solutios: (i) 0. +. +. + to tems Let, S =. +. +. + to tems = [. +. +. + to tems] Multiplyig ad dividig by 9 S = [.9 +.99 +.999 + to tems] 9 = [( ) + (.0) + (0..00) + to tems] 9 = 9 [(+++ tems) (0.+.0+.00+..to tems)] a =. =.0 0.. 0 a = 0 a( ) We use S = 0 0 S = 9 9 9 0 0 Solutio (ii) Let, S = (x + y) + (x + xy +y ) + (x 3 +x y+xy +y 3 ) +. to tem. Multiplyig ad dividig by (x y) S = [(x+y)(x-y)+(x-y)(x +xy+y )+(x-y) (x 3 +x y+xy +y 3 )+ (x y) S = [(x y )+ (x 3 y 3 )+ (x 4 y 4 )+. to tem] (x y)
Chapte 49 Sequece ad seies S = [(x + x 3 +x 4 +. to tem) (y +x 3 +y 4 +. to tem] (x y) a( ) We use S = x (x ) y (y ) S = (x y) x y Example 6: The sum of the fist 0 tems of a G.P. is equal to 44 times the sum of fist 5 tems. Fid commo atio. Solutio: Hee, = 0, = 5, =? a( ) So, S = 0 5 a( ) a( ) S 0 =, S 5 = By the Give coditio: S 0 = 44S 5 0 a( ) 5 a( ) = 44 0 = 44( 5 ) () ( ) 5 = 44( 5 ) ( 5 ) ( + 5 )= 44( 5 ) ( 5 )[ + 5-44] = 0 + 5 44 = 0 o 5 = 0 + 5 = 44 = 0 5 = 5 = 43 = 3 = which ot possible Example 7: Give = 6, = 3, S = 665 fid a. 44 Solutio: a( ) Fomula S = 6 a 665 3 44 3 >
Chapte 50 Sequece ad seies 64 a 79 = 3 665 665 a 44 43 665 43 a= x 44 665 7 a= 6 Example 8: If a ma deposits $ 00 at the begiig of each yea i a bak that pays 4 pecet compouded aually, how much will be to his cedit at the ed of 6 yeas? Solutio: The ma deposits $ 00 at the begiig of each yea. The bak pays 4% compouded iteest aually At the ed of fist yea the piciple amout o cedit becomes = 00(.04) At the begiig of secod yea the piciple amout o cedit is = 00 + 00 (.04) At the ed of secod yea the piciple amout o cedit becomes = 00(.04) + 00(.04) = 00(.04 +.04 ) So at the ed of 6 yeas the piciple amout o cedit becomes = 00 (.04 +.04 + sum upto 6 times) Coside,.04 +.04 + - - - - - - - - 6 tems. a =.04, =.04, ad = 6 By the fomula a( ) S = > 6.04(.04 ) S 6 =.04.04(.653 ) = 0.04.04 x 0.653 = 0.04 = 6.8983
Chapte 5 Sequece ad seies Hece at the ed of 6 yeas the cedit is = 00(6.8983) = $379.66 Execise.6 Q. Fid the sum of each of the followig seies: (i) - - - - - - - to 6 tems 3 9 (ii) x + x + x 3 - - - - - - - to 0 tems. (iii) - - - - - - - + 64 8 4 (iv) 3 + 3 + 3 3 + - - - - - - - - - + 3 Q. How may tems of the seies? 3 9 33 - - - - - - - amout to 3 4 48 Q3. Sum the seies. (i).3 +.33 +.333 + - - - - - - - to tems. (ii) 3 + 33 + 333 + - - - - - - - to tems. (iii) + ( + x) + (+ x+x ) +(+x+x +x 3 ) 3 + - - - to tems. Q4. What is the sum of the geometic seies fo which a =, = 5, l = a = 3? Q5. A ubbe ball is dopped fom a height of 4.8 dm. It cotiuously ebouds, each time eboudig 3 of the distace of the pecedig 4 fall. How much distace has it taveled whe it stikes the goud fo the sixth time? Q6. The fist tem of geometic pogessio is ad the 0th tem is 56, usig fomula fid sum of its tems. Q7. What is fist tem of a six tem G.P. i which the commo atio is 3 ad the sixth tem is 7 fid also the sum of the fist thee tems. Aswes.6. (i) 364 43 (ii) 0 x( x ) x (iii) 03/8
Chapte 5 Sequece ad seies (iv). = 6 3. (i) (iii) 3(3 ) 3 9 0 (ii) x( x ( x) x 4. 6 5. 6.76 6. 4085 0(0 ) 3 9 3(3 3 ) 7. 3; 3.4 Ifiite Geometic Sequece: A geometic sequece i which the umbe of tems ae ifiite is called as ifiite geometic sequece. Fo example: 4 8 6 (i),,,, - - - - - - 3 9 7 (ii), 4, 8, 6, 3, - - - - - - - Ifiite Seies: Coside a geometic sequece a, a, a, - - - - - - - to tems. Let S deote the sum of tems the S = a + a + a + - - - - - - to tems. a( ) Fomula S = Takig limit as o both sides ( ) limits = limit a = limit a = limit as, 0 Theefoe S = a limit a a 0
Chapte 53 Sequece ad seies a S = the fomula fo the sum of ifiite tems of G.P. Coveget Seies: A ifiite seies is said to be the coveget seies whe its sum teds to a fiite ad defiite limit. Fo example: - - - - - - - is a seies 3 3 6 Hee a = 3, = = 3 3 < a S = = 3 3 3 = x 4 3 3 Hece the seies is coveget. Diveget Seies: Whe the sum of a ifiite seies is ifiite, it is said to be the Diveget seies. Fo example: + 4 + 8 + 6 + 3 + - - - - - - - - Hee a =, =, > Theefoe we use fomula a( ) ( ) S = S = + limits = limit ( + ) + S = = as, + Hece the seies is a diveget seies.
Chapte 54 Sequece ad seies.4 Recuig Decimals: Whe we attempt to expess a commo factio such as 3 o as 8 4 as a decimal factio, the decimal always eithe temiates o ultimately epeats. 3 Thus = 0.375 (Decimal temiate) 8 4 = 0.363636 (Decimal epeats) We ca expess the ecuig decimal factio 0.36 (o 0.36 ) as a commo factio. The ba ( 0.36 ) meas that the umbes appeaig ude it ae epeated edlessly. i.e. 0.36 meas 0.363636 - - - - - - - Thus a o-temiatig decimal factio i which some digits ae epeated agai ad agai i the same ode i its decimal pats is called a ecuig decimal factio. Example : Fid the factio equivalet to the ecuig decimals 0.3. Solutio: Let S = 0.3 = 0.3 3 3 - - - - - - - - - - - = 0.3 + 0.0003 + 0.0000003 - - - - - - = 3 3 3 - - - - - - 000 000 000 000 000 000 3 3 3 = 3 6 9 - - - - - - 0 0 0 3 Hee a = 3 0, = 3 0 a S = 3 3 3 3 = 0 = 000 000 000 0 3 000 000 =
Chapte 55 Sequece ad seies 3 000 3 = x 000 999 999 4 = 333 Example : Fid the sum of ifiite geometic seies i which a = 8, =. Solutio: a = 8, = a Usig S = 8 8 S = = 8 8x 3 3 S = 56 3 Execise.7 Q. Fid the sum of the followig ifiite geometic seies (i) - - - - - - - 8 6 3 (ii) - - - - - - - Q. Fid the sum of the followig ifiite geometic seies (i) a = 3, = 3 (ii) a = 3, = 3 4 Q.3 Which of the followig seies ae (i) diveget (ii) coveget (i) + 4 + 6 + 64 + - - - - - - - - - (ii) 3 3 3 6 3 - - - - - - - 4 8
Chapte 56 Sequece ad seies (iii) 6 + + 4 + 48 + - - - - - - - - - Q.4 Fid the factios equivalet to the ecuig decimals. (i) 0.36 (ii).43 (iii) 0.836 Q.5 Fid the sum to ifiity of the seies + ( + k) + + k + k ) + ( + k + k + k 3 ) 3 + - - - - - - - ad k beig pope factio. Q.6 If y = x + x + x 3 + - - - - - - - ad if x is positive ad less tha y uity show that x = +y Q.7 What distace a ball tavel befoe comig to est if it is dopped fom a height of 6 dm ad afte each fall it ebouds of the 3 distace it fell. Q.8 The sum of a ifiite geometic seies i 5 ad the sum of the squaes of its tems is 45. Fid the seies. Aswes.7 Q. (i) S = (ii) S = 4 Q. (i) 9 (ii) Q.3 (i) Diveget (ii) Coveget (iii) Diveget Q.4 (i) 4 4 5 (ii) (iii) 99 6 Q.5 ( )( K) Q.7 30 dm. Q.8 0 0 5 - - - - - - - 5 9
Chapte 57 Sequece ad seies Summay. th tem of Geeal Tem of a Aithmetic pogessio. a = a + ( )d. Aithmetic meas betwee a ad b A = a + b 3. Sum of the Fist tems of a aithmetic seies. (i) (ii) S = [a + ( )d] S = (a + l) whe last tem is give. 4. Geeal o th tem of a G.P a = a - 5. Geometic meas betwee a ad b G = ab 6. Sum of tems of a Geometic Seies a( ) S = if < a( ) S = if > 7. Sum of a ifiite Geometic Seies S = a
Chapte 58 Sequece ad seies Shot questios Wite the shot aswes of the followig Q.: Defie a sequece. Q.: Q.3: Q.4: Q.5: Q.6: Q.7: Defie fiite sequece. Defie ifiite sequece. Defie commo diffeece. Wite the th tem of aithmetic pogessio. Fid the 7 th tem of A.P. i which the fist tem is 7 ad the commo diffeece is 3. Fid the 7 th tem of a AP, 4, 7,.. Q.8: Fid the sequece whose geeal tem i 4 +. Q.9: Defie a seies. Q.0: Wite the fomula to fid the sum of tem of a aithmetic sequece. Q.: Fid the sum of the seies 3 + + 9 + to 6 tems. Q.: Fid the sum of the seies 5 + 8 + + 4 +.to tems. Q.3: Defie aithmetic meas (AMs). Q.4: Fid the A.M. betwee 5 4 ad 5 + 4. Q.5: Defie a commo atio. Q.6: Wite the th tem of a geometic pogessios. Q.7: Fid the tem idicated i the followig G.P., 3 3, 3 6, 6 th tems. Q.8: wite dow the geometic sequece i which fist tem is ad the secod tem is 6 ad = 5. Q.9: Wite the fomula of sum of the fist tems of a geometic sequece fo < ad fo > Q.0: Defie geometic meas. Q.: Fid the G.M. betwee (i) 8 ad 7 (ii) 4 3, 43. Q.: Sum to 5 tem the seies + 3 +9 +. Q.3: Fid the sum of the followig seies: + 3 + 9 + to 6 tems. Q.4: Fid the sum of ifiite geometic seies i which a=8, =
Chapte 59 Sequece ad seies Q.5: Fid the sum of followig ifiite geometic seies + + +. Aswes Q6 a 7 = - Q7 a 7 = 9 Q8 5, 9, 3, Q 008 Q [7 + 3] Q4 5 Q7 (7) 5 Q8, -6, 8, -54, 6 Q (i) 4 (ii) 9 Q Q3 364 43 Q4 56 3 Q5 4 -
Chapte 60 Sequece ad seies Objective Type Questios Q. Each questios has fou possible aswes. Choose the coect.. 3. 4. 5. aswe ad ecicle it. The th tem of a A.P. whose st tem is a ad commo diffeece is d is: (a) a + ( + )d (b) a + ( + )d (c) a + ( )d (d) a + (d ) The th tem of a A.P., 4, 7, is: (a) 7 (b) 9 (c) (d) 3 If a, b, c ae i A.P. the: (a) b a = c b (b) b c a b (c) a + b = b + c (d) a b c a The 0th tem is 7, 7, 7, is: (a) 97 (b) 98 (c) 99 (d) 00 The sum of tems of a A.P. with a as st tem ad d as commo diffeece is: (a) [a + ( )d] (b) [a + ( )d] (c) [a + ( + )d] (d) [a ( )d] 6. Aithmetic mea betwee x 3ad x+ 3 is: (a) x (b) x (c) 3 (d) 3 7. If S = ( + + ) the its 4th tem will be: (a) (b) 40 (c) 4 (d) 0 8. Aithmetic mea betwee 7 ad 7 is: 7 (a) (b) 7 (c) 0 (d) 4 9. The sum of the seies + + 3 + + 00 is: (a) 00 (b) 5000 (c) 5050 (d) 500 0. The th tem of a G.P a, a, a, is: (a) a (b) a + (c) a - (d) a -. The 5th tem of a G.P,,, is: 4 (a) (b) (c) (d) 8 8 6 3
Chapte 6 Sequece ad seies. The 6th tem of G.P,, 4, is: (a) 4 (b) 4 (c) (d) 3. The G.M. betwee a ad b is: (a) ab (b) ab (c) ab (d) ab 4. If x, y, z ae i G.P. the: (a) y = x + z (b) y = xz (c) y = xz (d) z = xy 5. Geometic mea betwee 3 ad 7 is: (a) 9 (b) (c) 5 (d) 9 6. The sum of tems of a geometic seies: a+ a + a + ; < (a) - a a( ) (b) (c) + a a( ) (d) 7. The sum of 6 tems of the seies + + 4 + is: (a) 63 (b) 64 (c) 65 (d) 66 8. The sum of 5 tems of the seies + 4 : (a) 6 (b) (c) (d) 6 9. The sum of ifiite tems of a G.P. a, a, a, if < is: (a) a a( ) (b) (c) a - (d) Noe of these 0. The sum of ifiite geometic seies is: 3 9 3 (a) (b) 3 (c) (d) 3 Aswes. c. b 3. a 4. a 5. b 6. a 7. a 8. c 9. c 0. d. c. a 3. c 4. c 5. d 6. b 7. a 8. b 9. a 0. c 3