MISCELLANEOUS SEQUENCES & SERIES QUESTIONS

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Carmel Shelton
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1 MISCELLANEOUS SEQUENCES & SERIES QUESTIONS

2 Questio (***+) Evaluate the followig sum 30 r ( 2) 4r MP2-V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio with commo ratio r. Give further that A, 2B, C i that order are i arithmetic progressio, determie the possible values of r. MP2-X, r = 2 ± 3

3 Questio 3 (***+) A arithmetic series has commo differece 2. rd th th The 3, 6 ad 0 terms of the arithmetic series are the respective first three terms of a geometric series. Determie i ay order the first term of the arithmetic series ad the commo ratio of the geometric series. MP2-Z, a = 4, r = 4 3

4 Questio 4 (***+) Each of the terms of a arithmetic series is added to the correspodig terms of a geometric series, formig a ew series with first term 8 3 ad secod term 3 6. The commo differece of the arithmetic series is four times as large as the first term of the geometric series. The commo ratio of the geometric series is twice as large as the first term of the arithmetic series. Determie the possible values of the first term of the geometric series. MP2-M, 7 8 4

5 Questio 5 (****) Fid i simplified form, i terms of, the value of 2 r ( 3r 2)( ). SP-I, 3

6 Questio 6 (****) The st rd th, 3 ad term of a arithmetic progressio are the first three terms of a geometric progressio. It is further give that the sum of the first 3 terms of the arithmetic progressio is 260. Fid, i ay order, the commo ratio of the geometric progressio ad the first term ad commo differece of the arithmetic progressio. MP2-N, r = 4

7 Questio 7 (****) d rd th The 2, 3 ad 9 term of a arithmetic progressio are three cosecutive terms of a geometric progressio. Fid the commo ratio of the geometric progressio. MP2-R, r = 3 4

8 Questio 8 (****+) It is give that xr = 2 ad = r xr x 3 2 r =. 2 2 ( ) Determie, i terms of, the value of ( x ) 2 r +. MP2-O, ( ) 2 x r + = 8

9 Questio 9 (****+) It is give that Fid the value of ( r 0) = 200 ad ( r ) 2 0 = r. MP2-Q, 20 2 r = 8800

10 Questio 0 (****+) The sum of the first 2 terms of a arithmetic progressio is 40. The sum of the first 4 terms of the same arithmetic progressio is 30. a) Determie the sum of the first 5 terms of the arithmetic progressio. The sum of the first 2 terms of a geometric progressio is 40. The sum of the first 4 terms of the same geometric progressio is 30. b) Fid the two possible values of the sum of the first 5 terms of the geometric progressio. MP2-P, S = = 93.75, S 4 5 = 2 or S5 = 275

11 Questio (*****) It is give that ur =, 8 where u is the th term of a sequece. Fid a simplified expressio for u. SPX-C, u = 9 5

12 Questio 2 (*****) It is give that 2 + ur = ( 3 2) 2, where u is the th term of a sequece. Fid a simplified expressio for u. u = SP-B, ( )

13 Questio 3 (*****) It is give that + ur = , where u is the th term of a sequece. Show clearly that u = Au + Bu, where A ad B are itegers to be foud. SP-P, u + 2 = 8u + 2u

14 Questio 4 (*****) Fid i exact simplified form a exact expressio for the sum of the first terms of the followig series SP-N, S = 8

15 Questio 5 (*****) The first three terms of a geometric progressio are the respective d ad 2 term of a arithmetic progressio. Determie the commo ratio of the geometric progressio. th 7 term, th 4 term, MP2-S, r = 2 3

16 Questio 6 (*****) A factory gets permissio to dispose, at the start of every day, 600 kg of waste ito a stream of water. The ruig stream removes 40% of the ay waste preset, by the ed of the day. Determie a simplified expressio for the amout of waste preset i the stream at the th ed of the day. MP2-T, u = ( ) 5

17 Questio 7 (*****) Express the sum of the first terms of the above series i sigma otatio. You are ot required to sum the series. S r ( ) = 0 3

18 Questio 8 (*****) A rectagle has perimeter P ad area A. Show that A f ( P), where f ( P ) is a simplified expressio to be foud. SP-K, A P 6 2

19 Questio 9 (*****) A sequece of positive itegers is geerated by the formula 3 2 u = , N. Determie the largest value of, such that u > u +. SP-J, = 9

20 Questio 20 (*****) The geometric mea of two positive umbers a ad b is deoted by G. The arithmetic mea of a ad b is deoted by A. Give further that the ratio : G = 4 : 5, determie the ratio betwee a ad b. A SP-P, 4:

21 Questio 2 (*****) A fuctio is defied as [ x] { the greatest iteger less or equal to x}. The fuctio f is defied as 3 3 f ( ) = , N. Determie the value of 82 f ( ) =. SP-V, 5797

22 Questio 22 (*****) The sum to ifiity S of the coverget geometric series is give by S = + x + x + x + x +..., x <, By itegratig the above equatio betwee suitable limits, or otherwise, fid 2 r r. You may assume that itegratio ad summatio commute. SP-C, l 2

23 Questio 23 (*****) Evaluate the followig expressio k= k r. SPX-B, 2

24 Questio 24 (*****) It is give that L is the sum to ifiity of the followig coverget series r!. 0 Use this fact to fid, i terms of L, the sum to ifiity of this coverget series 3 r. r! SP-A, 5L

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,