RI Sequences and Series C7A Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ____________________________________ Tutorial 7A: Sequences and Series Page 1 of 3 Tutorial 7A: Sequences and Series Section A (Basic Questions) 1 Write down the series given by the summation notation, showing the first three terms and the last two terms whenever possible. (a) 8 1 4 4r r r   (b) 30 4 (5 3 ) r r   (c)  15 5 3 j m j   , where m is a constant 2 For each of the following series, write down the rth term and express the series using the summation notation. (a) 3 6 9 ... 369  (b) ( 4) ( 1) (2) ... (53)   (c) 114 2 1 ... 24  (d) 22 22 22 1234 78 ...2345 89  3 Justify whether each of the following statements is true or false. (a)  22 1 44 10 1 nn rr rr    (b)  1 111 2 n r nn   (c) 2 2 10 nn rr rr   (d) 33 55 10 nn rr rr   4 It is given that 1 52 n n r r un   , where ru is an expression in terms of r. Find the value of (i) 10 1 r r u   (ii) 10 5 r r u   (iii) 10u

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ____________________________________ Tutorial 7A: Sequences and Series Page 2 of 3 Section B (Discussion Questions) 1 Use the following standard results 1 n r an a   , where a is a constant, 1 1 (1 )2 n r rn n   , 2 1 1 (1 ) ( 21 )6 n r rn n n   and 32 2 1 1 (1 )4 n r rn n   to evaluate each of the following sum. (a) 15 2 0 (4 ) r r   (b) 11 2 4 (3 ) r rr   (c) 10 1 (2 ) r rr   For each part, use the graphing calculator to verify that your answer is correct. 2 A sequence is such that 1up , where p is a constant, and 1 25nnuu  , for n > 0. (i) Describe how the sequence behaves when (A) p = 7, [1] (B) p = 5. [1] (ii) Find the value of p for which 5 101u  . [2] [ ( i i ) 11] 3 It is given that 1 11 111 n r rr n  . (a) Find  1 5 1 1 n r rr    . (b) Give a reason why the series in part (a) is convergent and state the value of 5 1 1r rr    . [(a) 11 52 n  (b) 1 5 ] 4 A sequence 123, , , ...uuu is such that 1 2u  , 1 35 n n uu   for all 1n , where k is a constant. (i) Write down the values of 2u , 3u and 4u . [2] (ii) Given that the sequence converges to l, find the value of l. [1] [(i) 23 4 2.6, 3.52, 3.704uu u   , (ii) 3.75l  ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ____________________________________ Tutorial 7A: Sequences and Series Page 3 of 3 5 The sum, nS , of the first n terms of a sequence 12 3, , ,uuu  is given by   2 nSn A n B , where A and B are non-zero constants. (i) Find an expression for nu in terms of A, B and n, simplifying your answer. [3] (ii) It is also given that the second term is 6 and the sixth term is 218. Find the values of A and B. [2] (iii) Determine if the sum to infinity exists. [1] [(i)     233 1 2 1nuA n n B n    , (ii) 3, 5AB  , (iii) No] 6 For this question, you may use the results   2 1 12 1 6 n r nn nr   and 3 22 1 4 1 n r nnr   . (i) Find  2 1 12 n r rr   in terms of n. [2] (ii) Find  2 1 1 n r rr   in terms of n. Hence find  1 2 2 1 n r rr    in terms of n. [5] (iii) Without using a graphing calculator, find the sum of the series      4 25 5 36 6(49) 7(64) 59 3600   . [3] [(i)   21 13 16 nn n n  (ii)    1 13 2 112 nn n n ,    1 11 3 2 412 nn n n   (iii)  108836] 7 It is given that  232 1 1 14 n r rn n   . (a) Find  2 3 1 n rn r   where 1n . Express your answer in the form   22 21 12 1na n na   , where a is a constant to be determined. [4] (b) Find 333 3 31 3 5 ... (2 3) (2 1) kk    , where k is a positive integer. [2] [(a) a = 4 (b) 422kk ] THE END