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Transcription

1 ARITHMETIC PROGRESSIONS (A) Mai Cocepts ad Results A arithmetic progressio (AP) is a list of umbers i which each term is obtaied by addig a fixed umber d to the precedig term, except the first term a. The fixed umber d is called its commo differece. The geeral form of a AP is a, a + d, a + d, a + 3d,... I the list of umbers a, a, a 3,... if the differeces a a, a 3 a, a 4 a 3,... give the same value, i.e., if a k + a k is the same for differet values of k, the the give list of umbers is a AP. The th term a (or the geeral term) of a AP is a = a + ( ) d, where a is the first term ad d is the commo differece. Note that a a. The sum S of the first terms of a AP is give by S = [a + ( ) d] If l is the last term of a AP of terms, the the sum of all the terms ca also be give by S = [a + l] Sometimes S is also deoted by S. CHAPTER 5 ot to be republished

2 ARITHMETIC PROGRESSIONS 45 If S is the sum of the first terms of a AP, the its th term a is give by a = S S (B) Multiple Choice Questios Choose the correct aswer from the give four optios: Sample Questio : The 0 th term of the AP: 5, 8,, 4,... is (A) 3 (B) 35 (C) 38 (D) 85 Solutio : Aswer (A) Sample Questio : I a AP if a = 7., d = 3.6, a = 7., the is (A) (B) 3 (C) 4 (D) 5 Solutio : Aswer (D) EXERCISE 5. Choose the correct aswer from the give four optios:. I a AP, if d = 4, = 7, a = 4, the a is (A) 6 (B) 7 (C) 0 (D) 8. I a AP, if a = 3.5, d = 0, = 0, the a will be (A) 0 (B) 3.5 (C) 03.5 (D) The list of umbers 0, 6,,,... is (A) a AP with d = 6 (B) a AP with d = 4 (C) a AP with d = 4 (D) ot a AP ot to be republished 4. The th term of the AP: 5, 5, 0, 5,...is (A) 0 (B) 0 (C) 30 (D) The first four terms of a AP, whose first term is ad the commo differece is, are

3 46 EXEMPLAR PROBLEMS (A), 0,, 4 (B), 4, 8, 6 (C), 4, 6, 8 (D), 4, 8, 6 6. The st term of the AP whose first two terms are 3 ad 4 is (A) 7 (B) 37 (C) 43 (D) If the d term of a AP is 3 ad the 5 th term is 5, what is its 7 th term? (A) 30 (B) 33 (C) 37 (D) Which term of the AP:, 4, 63, 84,... is 0? (A) 9 th (B) 0 th (C) th (D) th 9. If the commo differece of a AP is 5, the what is a8 a 3? (A) 5 (B) 0 (C) 5 (D) What is the commo differece of a AP i which a 8 a 4 = 3? (A) 8 (B) 8 (C) 4 (D) 4. Two APs have the same commo differece. The first term of oe of these is ad that of the other is 8. The the differece betwee their 4 th terms is (A) (B) 8 (C) 7 (D) 9. If 7 times the 7 th term of a AP is equal to times its th term, the its 8th term will be (A) 7 (B) (C) 8 (D) 0 3. The 4 th term from the ed of the AP:, 8, 5,..., 49 is (A) 37 (B) 40 (C) 43 (D) The famous mathematicia associated with fidig the sum of the first 00 atural umbers is ot to be republished (A) Pythagoras (B) Newto (C) Gauss (D) Euclid 5. If the first term of a AP is 5 ad the commo differece is, the the sum of the first 6 terms is (A) 0 (B) 5 (C) 6 (D) 5

4 ARITHMETIC PROGRESSIONS The sum of first 6 terms of the AP: 0, 6,,... is (A) 30 (B) 30 (C) 35 (D) I a AP if a =, a = 0 ad S = 399, the is (A) 9 (B) (C) 38 (D) 4 8. The sum of first five multiples of 3 is (A) 45 (B) 55 (C) 65 (D) 75 (C) Short Aswer Questios with Reasoig Sample Questio : I the AP: 0, 5, 0, 5,... the commo differece d is equal to 5. Justify whether the above statemet is true or false. Solutio : a a = 5 0 = 5 a 3 a = 0 5 = 5 a 4 a 3 = 5 0 = 5 Although the give list of umbers forms a AP, it is with d = 5 ad ot with d = 5 So, the give statemet is false. Sample Questio : Divya deposited Rs 000 at compoud iterest at the rate of 0% per aum. The amouts at the ed of first year, secod year, third year,..., form a AP. Justify your aswer. Solutio : Amout at the ed of the st year = Rs 00 Amout at the ed of the d year = Rs 0 Amout at the ed of 3rd year = Rs 33 ad so o. So, the amout (i Rs) at the ed of st year, d year, 3rd year,... are 00, 0, 33,... Here, a a = 0 As, ot to be republished a 3 a = a a a 3 a, it does ot form a AP.

5 48 EXEMPLAR PROBLEMS Sample Questio 3: The th term of a AP caot be +. Justify your aswer. Solutio : Here, a So, a a 5 a List of umbers becomes, 5, 0,... Here, 5 Alterative Solutio : 0 5, so it does ot form a AP. We kow that i a AP, d a a Here, a So, a a ( ) ( ) As = a a depeds upo, d is ot a fixed umber. So, a caot be the th term of a AP. Alterative Solutio : We kow that i a AP ot to be republished ( ) a = a+ d. We observe that a is a liear polyomial i. Here, a is ot a liear polyomial i. So, it caot be the th term of a AP.

6 ARITHMETIC PROGRESSIONS 49 EXERCISE 5.. Which of the followig form a AP? Justify your aswer. (i),,,,... (ii) 0,, 0,,... (iii),,,, 3, 3,... (iv),, 33,... (v), 3, 4,... (vi),, 3, 4,... (vii) 3,, 7, 48,.... Justify whether it is true to say that, a a = a 3 a. 3,, 5,... forms a AP as 3. For the AP: 3, 7,,..., ca we fid directly a 30 a 0 without actually fidig a 30 ad a 0? Give reasos for your aswer. 4. Two APs have the same commo differece. The first term of oe AP is ad that of the other is 7. The differece betwee their 0 th terms is the same as the differece betwee their st terms, which is the same as the differece betwee ay two correspodig terms. Why? 5. Is 0 a term of the AP: 3, 8, 5,...? Justify your aswer. 6. The taxi fare after each km, whe the fare is Rs 5 for the first km ad Rs 8 for each additioal km, does ot form a AP as the total fare (i Rs) after each km is 5, 8, 8, 8,... ot to be republished Is the statemet true? Give reasos. 7. I which of the followig situatios, do the lists of umbers ivolved form a AP? Give reasos for your aswers. (i) The fee charged from a studet every moth by a school for the whole sessio, whe the mothly fee is Rs 400.

7 50 EXEMPLAR PROBLEMS (ii) (iii) (iv) The fee charged every moth by a school from Classes I to XII, whe the mothly fee for Class I is Rs 50, ad it icreases by Rs 50 for the ext higher class. The amout of moey i the accout of Varu at the ed of every year whe Rs 000 is deposited at simple iterest of 0% per aum. The umber of bacteria i a certai food item after each secod, whe they double i every secod. 8. Justify whether it is true to say that the followig are the th terms of a AP. (i) 3 (ii) 3 +5 (iii) ++ (D) Short Aswer Questios Sample Questio : If the umbers, 4 ad 5 + are i AP, fid the value of. Solutio : As, 4, 5 + are i AP, so (4 ) ( ) = (5 + ) (4 ) i.e, 3 + = + 3 i.e, = Sample Questio : Fid the value of the middle most term (s) of the AP :, 7, 3,..., 49. Solutio : Here, a =, d = 7 ( ) = 4, a = 49 We have a = a + ( ) d So, 49 = + ( ) 4 i.e., 60 = ( ) 4 i.e., = 6 As is a eve umber, there will be two middle terms which are ot to be republished 6 6 th ad + th, i.e., the 8 th term ad the 9 th term.

8 ARITHMETIC PROGRESSIONS 5 a 8 = a + 7d = = 7 a 9 = a + 8d = = So, the values of the two middle most terms are 7 ad, respectively. Sample Questio 3: The sum of the first three terms of a AP is 33. If the product of the first ad the third term exceeds the secod term by 9, fid the AP. Solutio : Let the three terms i AP be a d, a, a + d. So, a d + a + a + d = 33 or a = Also, (a d) (a + d) = a + 9 i.e., a d = a + 9 i.e., d = + 9 i.e., d = 8 i.e., d = ± 9 So there will be two APs ad they are :,, 0,... ad 0,,,... EXERCISE 5.3. Match the APs give i colum A with suitable commo differeces give i colum B. Colum A Colum B (A ),, 6, 0,... (B ) (A ) a = 8, = 0, a = 0 (B ) 5 ot to be republished (A 3 ) a = 0, a 0 = 6 (B 3 ) 4 (A 4 ) a = 3, a 4 =3 (B 4 ) 4 (B 5 ) (B 6 ) (B 7 ) 5 3

9 5 EXEMPLAR PROBLEMS. Verify that each of the followig is a AP, ad the write its ext three terms. (i) 0, 4,, 3 4,... (ii) 5, 4 3, 3 3, 4,... (iii) 3, 3, 3 3,... (iv) a + b, (a + ) + b, (a + ) + (b + ),... (v) a, a +, 3a +, 4a + 3, Write the first three terms of the APs whe a ad d are as give below: (i) a =, d = 6 (ii) a = 5, d = 3 (iii) a =, d = 4. Fid a, b ad c such that the followig umbers are i AP: a, 7, b, 3, c. 5. Determie the AP whose fifth term is 9 ad the differece of the eighth term from the thirteeth term is The 6 th, th ad the last term of a AP are 0, 3 ad commo differece ad the umber of terms., respectively. Fid the 5 7. The sum of the 5 th ad the 7 th terms of a AP is 5 ad the 0 th term is 46. Fid the AP. ot to be republished 8. Fid the 0 th term of the AP whose 7 th term is 4 less tha the th term, first term beig. 9. If the 9 th term of a AP is zero, prove that its 9 th term is twice its 9 th term. 0. Fid whether 55 is a term of the AP: 7, 0, 3,--- or ot. If yes, fid which term it is.

10 ARITHMETIC PROGRESSIONS 53. Determie k so that k + 4k + 8, k + 3k + 6, 3k + 4k + 4 are three cosecutive terms of a AP.. Split 07 ito three parts such that these are i AP ad the product of the two smaller parts is The agles of a triagle are i AP. The greatest agle is twice the least. Fid all the agles of the triagle. 4. If the th terms of the two APs: 9, 7, 5,... ad 4,, 8,... are the same, fid the value of. Also fid that term. 5. If sum of the 3 rd ad the 8 th terms of a AP is 7 ad the sum of the 7 th ad the 4 th terms is 3, fid the 0 th term. 6. Fid the th term from the ed of the AP:, 4, 6,..., Which term of the AP: 53, 48, 43,... is the first egative term? 8. How may umbers lie betwee 0 ad 300, which whe divided by 4 leave a remaider 3? 9. Fid the sum of the two middle most terms of the AP: 4,, 3,..., The first term of a AP is 5 ad the last term is 45. If the sum of the terms of the AP is 0, the fid the umber of terms ad the commo differece.. Fid the sum: (i) + ( ) + ( 5) + ( 8) ( 36) (ii) (iii) upto terms a b 3 a b 5 a 3b to terms. a+ b a+ b a+ b ot to be republished. Which term of the AP:, 7,,... will be 77? Fid the sum of this AP upto the term If a = 3 4, show that a, a, a 3,... form a AP. Also fid S I a AP, if S = (4 + ), fid the AP.

11 54 EXEMPLAR PROBLEMS 5. I a AP, if S = ad a k = 64, fid the value of k. 6. If S deotes the sum of first terms of a AP, prove that S = 3(S 8 S 4 ) 7. Fid the sum of first 7 terms of a AP whose 4 th ad 9 th terms are 5 ad 30 respectively. 8. If sum of first 6 terms of a AP is 36 ad that of the first 6 terms is 56, fid the sum of first 0 terms. 9. Fid the sum of all the terms of a AP whose middle most term is Fid the sum of last te terms of the AP: 8, 0,,---, Fid the sum of first seve umbers which are multiples of as well as of 9. [Hit: Take the LCM of ad 9] 3. How may terms of the AP: 5, 3,,--- are eeded to make the sum 55? Explai the reaso for double aswer. 33. The sum of the first terms of a AP whose first term is 8 ad the commo differece is 0 is equal to the sum of first terms of aother AP whose first term is 30 ad the commo differece is 8. Fid. 34. Kaika was give her pocket moey o Ja st, 008. She puts Re o Day, Rs o Day, Rs 3 o Day 3, ad cotiued doig so till the ed of the moth, from this moey ito her piggy bak. She also spet Rs 04 of her pocket moey, ad foud that at the ed of the moth she still had Rs 00 with her. How much was her pocket moey for the moth? 35. Yasmee saves Rs 3 durig the first moth, Rs 36 i the secod moth ad Rs 40 i the third moth. If she cotiues to save i this maer, i how may moths will she save Rs 000? ot to be republished (E) Log Aswer Questios Sample Questio : The sum of four cosecutive umbers i a AP is 3 ad the ratio of the product of the first ad the last terms to the product of the two middle terms is 7 : 5. Fid the umbers. Solutio: Let the four cosecutive umbers i AP be a 3d, a d, a + d, a + 3d.

12 ARITHMETIC PROGRESSIONS 55 So, a 3d + a d + a + d + a + 3d = 3 or 4a = 3 or a = 8 Also, or, a 3d a 3d 7 a d a d 5 a 9d 7 a d 5 or, 5 a 35 d = 7a 7 d or, 8 a 8 d = 0 or, d = or, d = ± So, whe a = 8, d =, the umbers are, 6, 0, 4. Sample Questio : Solve the equatio : Solutio : x =87 Here,, 4, 7, 0,..., x form a AP with a =, d = 3, a = x We have, a = a + ( )d So, x = + ( ) 3 = 3 ot to be republished Also, S = ( a l ) So, 87 = ( x)

13 56 EXEMPLAR PROBLEMS = ( 3 ) or, 574 = (3 ) or, = 0 Therefore, = ± 83 8 = = 84, = 4, 4 3 As caot be egative, so = 4 Therefore, x = 3 = 3 4 = 40. Alterative solutio: Here,, 4, 7, 0,... x form a AP with a =, d = 3, S = 87 We have, S= a d So, 87 3 or, 574 (3 ) or, ot to be republished Now proceed as above. EXERCISE 5.4. The sum of the first five terms of a AP ad the sum of the first seve terms of the same AP is 67. If the sum of the first te terms of this AP is 35, fid the sum of its first twety terms.

14 ARITHMETIC PROGRESSIONS 57. Fid the (i) sum of those itegers betwee ad 500 which are multiples of as well as of 5. (ii) sum of those itegers from to 500 which are multiples of as well as of 5. (iii) sum of those itegers from to 500 which are multiples of or 5. [Hit (iii) : These umbers will be : multiples of + multiples of 5 multiples of as well as of 5 ] 3. The eighth term of a AP is half its secod term ad the eleveth term exceeds oe third of its fourth term by. Fid the 5 th term. 4. A AP cosists of 37 terms. The sum of the three middle most terms is 5 ad the sum of the last three is 49. Fid the AP. 5. Fid the sum of the itegers betwee 00 ad 00 that are (i) divisible by 9 (ii) ot divisible by 9 [Hit (ii) : These umbers will be : Total umbers Total umbers divisible by 9] 6. The ratio of the th term to the 8 th term of a AP is : 3. Fid the ratio of the 5 th term to the st term, ad also the ratio of the sum of the first five terms to the sum of the first terms. 7. Show that the sum of a AP whose first term is a, the secod term b ad the last term c, is equal to 8. Solve the equatio a c b c a b a ot to be republished 4 + ( ) x = Jaspal Sigh repays his total loa of Rs 8000 by payig every moth startig with the first istalmet of Rs 000. If he icreases the istalmet by Rs 00 every moth, what amout will be paid by him i the 30 th istalmet? What amout of loa does he still have to pay after the 30 th istalmet?

15 58 EXEMPLAR PROBLEMS 0. The studets of a school decided to beautify the school o the Aual Day by fixig colourful flags o the straight passage of the school. They have 7 flags to be fixed at itervals of every m. The flags are stored at the positio of the middle most flag. Ruchi was give the resposibility of placig the flags. Ruchi kept her books where the flags were stored. She could carry oly oe flag at a time. How much distace did she cover i completig this job ad returig back to collect her books? What is the maximum distace she travelled carryig a flag? ot to be republished