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1 SERIES

2 SUMMATIONS BY STANDARD RESULTS

3 Questio (**) Use stadard results o summatios to fid the value of 48 ( r )( 3r ). 36 FP-B, Questio (**+) Fid, i fully simplified factorized form, a expressio for the sum of the first terms of the followig series. ( 5 3) + ( 7) + ( 7 ) + ( 3 5 ) +... FP-C, ( 8 + 7)

4 Questio 3 (***) It is give that ( 3r + a)( r + ) ( + )( + b ). Determie the values of each of the costats a ad b. FP-D, a =, b = 3

5 Questio 4 (**) Use stadard results o summatios to show that r ( r + )( r + 5) = ( + )( + )( + 7). 4 proof Questio 5 (**) Use stadard results o summatios to show that ( r )( r + ) = ( )( + 5). 6 proof Questio 6 (**) Use stadard results o summatios to show that r ( r ) = ( )( + )( 3 + ). proof Questio 7 (**) Use stadard results o summatios to show that r ( 8r + ) = ( + )( + ). proof

6 Questio 8 (**) Use stadard results o summatios to show that r ( r )( r + ) = ( )( + )( + ). 4 proof Questio 9 (**) Use stadard results o summatios to show that r ( r + )( r + 4) = ( + )( + )( 3 + 7). proof Questio 0 (**) Use stadard results o summatios to show that r ( r + ) = ( + )( + )( 3 + 5). proof

7 Questio (**) Show by usig stadard summatio results that. 6 a) ( r + )( r + 5) = ( + 7)( + 7) = b) ( r )( r ) proof Questio (**) Show by usig stadard summatio results that. 3 a) ( k k ) = ( + )( ) k= = 049. b) ( k k ) k= 0 proof

8 Questio 3 (**+) Fid, i fully factorized form, a expressio for the sum ( 3r ). FP-P, ( 4 + 3) Questio 4 (***) Show clearly that = +. 6 ( 3) ( 4) ( 3 4 )... ( )( ) ( 5)( ) proof

9 Questio 5 (***) Use stadard results o summatios to show that ( 3r + r ) ( + ). FP-J, proof Questio 6 (***) Use stadard results o summatios to show that k ( ) k ( k + )( 6k + ). = FP-K, proof

10 Questio 7 (***) Use stadard results o summatios to fid the value of the followig sum. 6 ( k )( k + ). k= FP-L, 600

11 Questio 8 (***+) Fid the sum of the first terms of the series Express the aswer as a product of liear factors. FP-R, proof

12 Questio 9 (***+) By usig stadard results, show that 4 ( r ) ( 84 ). + SYNF-B, proof

13 Questio 0 (***+) Fid, i fully factorized form, a expressio for the followig sum. 3 ( r r ). 3 3 FP-Q, ( r r) = ( 5 4)( + ) 4

14 Questio (***+) It is thought that for some values of the costats p ad q that r ( r + p) ( + )( + )( 3 + q ). Use a detailed method to show that there exist o such values of p ad q. FP-F, proof

15 Questio (****) Use stadard results o summatios to solve the followig equatio. k 3 ( r ) = FP-E, k = 4

16 Questio 3 (****) It is give that 3 ( Ar + Br + Cr) = ( + )( + )( 4 5 ). Use a detailed method to fid the value of each of the iteger costats, A, B ad C. FP-A, A = 6, B = 3, C = 9

17 Questio 4 (****) Show by a detailed method that r ( r 3r ) + + = ( ). 0 FP-N, proof

18 Questio 5 (****) The sum, S, of the first terms of a series whose geeral term is deoted by u is give by the followig expressio. ( )( ) S = + +. a) Fid the first term of the series. b) Show clearly that i. u = ( + )( 4 ) r. + ii. u = 3 ( + )( 5 + ) FP-O, u = 6

19 Questio 6 (****) 3 ( ar + br + c) , where a, b ad c are iteger costats. Determie the value of a, b ad c. a = 3, b = 7, c =

20 Questio 7 (****) The variace Var ( ) of the first atural umbers is give by. Var ( ) = r r Determie the a simplified expressio Var ( ) ad hece evaluate Var ( 6 ). ( ) ( ) Var Var 6 = 30 =, ( )

21 Questio 8 (****), N. 3 ( ) f = r r a) Use stadard summatio results to fid a fully factorized expressio for f ( ). b) Hece solve the equatio 0 3 r r + 6k r + k = 70 5 = 4 + +, k =, k = 5 FP-M, f ( ) ( )( )( )

22 Questio 9 (****+) It is give that Fid the value of 0 0 ( r 0) = 00 ad ( r ) 0 = r. FP-U, 0 r = 8800

23 Questio 30 (****+) ( r + a)( r + b) ( )( + 4), 3 where a ad b are iteger costats. Use a clear algebraic method to determie the value of a ad the value of b. FP-V, ad (i ay order)

24 Questio 3 (****+) By usig a algebraic method, fid the value of FP-W, 5000

25 Questio 3 (****+) Show clearly that = SPX-A, proof

26 Questio 33 (****+) The positive iteger fuctios f ad g are defied as Evaluate 3 ( ) = r ad g ( ) = + ( r + ) f. 39 f ( ) g ( ). = SPX-E, 535

27 Questio 34 (*****) Fid the sum of the first 6 terms of the followig series SPX-M, 446

28 Questio 35 (*****) The fuctio f is defied for N as ( ) ( ) ( ) ( ) ( ) f Determie a simplified expressio for the sum of f ( ), givig the fial aswer i fully factorized form. FP-S, f ( ) = ( + )( + )

29 Questio 36 (*****) Use a algebraic method justifyig each step, to fid the greatest value of k, k N, which satisfies the followig iequality. 80 r > log r ( 6) k+ 8 SPX-G, k = 48

30 Questio 37 (*****) Use algebra to fid the sum of the first 00 terms of the followig sequece. 7,, 9, 8, 39, 5, 67, 84, 03,... = + + FP-T, f ( ) ( )( )

31 SUMMATIONS BY THE METHOD OF DIFFERENCES

32 Questio (**) ( ) f r a) Express f ( r ) ito partial fractios = 5 ( 5r )( 5r + 4), r N b) Hece show that 5 f ( r) =. 4( 5 + 4) ( ) f r = 5r 5r + 4

33 Questio (**) a) Show carefully that r + = r r r r ( + ) ( + ) b) Hece use the method of differeces to fid. r + r ( r + ). ( + )

34 Questio 3 (**) a) Show carefully that r =. r! r! r! ( + ) ( + ) b) Hece fid r!. ( r + )! ( + )

35 Questio 4 (**+) ( ) f r = r r ( + ), r N a) Express f ( r ) ito partial fractios. b) Hece show that f ( r) =. 984 ( ) f r = r ( r + )

36 Questio 5 (**+) ( ) f r = ( r + )( r + 3), r N a) Express f ( r ) ito partial fractios b) Use the method of differeces to fid c) Hece evaluate f ( r). ( ) f r = r r 3 f ( r) , ( ) 5 f r =, f ( r) = 9 90

37 Questio 6 (***) a) Simplify r r r r ( + ) ( + )( + ) ito a sigle fractio. b) Hece show that 0 5 =. r ( r + )( r + ) 46 FP-N, ( + )( r + ) r r

38 Questio 7 (***) ( ) ( ) ( ) f r r r + r r, r N. a) Simplify f ( r ) as far as possible. b) Use the method of differeces to show that 0 3 r = f r = 4r FP-M, ( ) 3

39 Questio 8 (***) ( ) f r = r r ( + ), r N. a) Express f ( r ) i partial fractios. b) Hece prove, by the method of differeces, that ( A + B) f ( r) =, 4( + )( + ) where A ad B are costats to be foud. FP-O, A = 3, B = 5

40 Questio 9 (***+) Use the method of differeces to show that ( )( ) ( + 3) ( )( ) = FP-R, proof

41 Questio 0 (***+) ur = r ( r + )( 4r + ), r N. 6 a) Simplify ur ur as far as possible. b) By usig the method of differeces, or otherwise, fid the sum of the first 00 terms of the followig series. ( 5) + ( 7) + ( 3 9) + ( 4 ) +... FP-L, r ( r + 3), 69850

42 Questio (***+) ( ) f r = ( r + )( r ), r N. a) Express f ( r ) ito partial fractios. b) Hece show that c) State the value of 3 =. r 4 ( + ) r FP-Q, f ( r) = ( r ) ( r + ), 3 4

43 Questio (***+) ( ) f r = r r ( + )( r + ), r N. a) Express f ( r ) ito partial fractios. b) Hece show that f ( r) =. ( + )( + ) c) Fid the value of the coverget ifiite sum FP-K, f ( r) = + r r + r +, 60

44 Questio 3 (***+) Use the method of differeces to show that = ( )( ) ( )( ) FP-P, proof

45 Questio 4 (****) ( ) f r = r + + r, r 0. a) Ratioalize the deomiator of f ( r ). b) Fid a expressio for c) Show clearly that ( ) f r 48 f ( r). ( ) f r = 3 + r + r =, f ( r) = ( )

46 Questio 5 (****+) Use partial fractios to sum the followig series = You may assume that the series coverges.

47 Questio 6 (*****) Determie the exact value of the followig sum = SPX-P, 499 0

48 Questio 7 (*****) f ( x, ) By observig the simplificatio of = fid a simplified expressio for f ( x, ). ( x ) r, x R, N. r ( x )( x ) ( x )( x ) r+ SPX-H, f ( x, ) = x x x ( )( )

49 Questio 8 (*****) Determie, i terms of k ad, a simplified expressio for r ( k ) r. r ( r ) k SPX-J, k k

50 Questio 9 (*****) Determie the value of the followig ifiite coverget sum. r 4r. r ( r ) 3 SPX-N, 3

51 Questio 0 (*****) Determie a simplified expressio, i the form l f ( ), for the followig sum. N r dx. x SPX-C, 3 l N N N ( + )

52 Questio (*****) Show, by a detailed method, that = A + B r, 50 where A ad B are costats to be foud. SPX-R, A = 5, B =

53 Questio (*****) , Show, by a detailed method, that the sum of the first 40 terms of this series show above is SPX-T, proof

54 Questio 3 (*****) By cosiderig the simplificatio of ( ) ( ) arcta + arcta, determie the exact value of arcta. = SP-U, π 4

55 Questio 4 (*****) ( ) ( ) ( ) ( ) ( ) S =! + 5! + 0 3! + 7 4! ! Use a appropriate method to show that S ( ) = +! SPX-V, proof

56 Questio 5 (*****) By cosiderig the trigoometric idetity for ta ( A B), with A arcta ( ) B = arcta( ), sum the followig series = + ad arcta. + + = You may assume the series coverges. SPX-L, π 4

57 Questio 6 (*****) Determie, i terms of, a simplified expressio r + 9r + 9, ( r + 5 )! ad hece, or otherwise, deduce the value of r + 7r +. ( r + 4 )! SPX-D, r + 9r =, ( r + 5 )! 6 ( + 5 )! r + 7r + 5 = 4! 4 ( r + )

58 Questio 7 (*****) A sequece is defied as r ur + = ur + 4 r + r +, u = 0, r N. Determie the exact value of u SPX-C, u 6 = 366

59 Questio 8 (*****) Fid the value of Hit: Express 4 si θ i terms of 0 r ( π ) si. r 4 si θ ad 4 si θ oly. SP-H,

60 Questio 9 (*****) Fid the sum to ifiity of the followig coverget series ! 5 3! 6 4! 7 5! 8 6! FP-T, 6

61 Questio 30 (*****) Evaluate the followig expressio k= k r. SPX-B,

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