RI Sequences and Series C7A Add Prac (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________________________________________ Additional Practice Questions for Chapter 7A: Sequences and Series Page 1 of 5 Additional Practice Questions for Chapter 7A: Sequences and Series 1 Evaluate the following sum. (a)   1 3 5 n r n r    (b) 2 1 ( )( 1) n r n r n r     (c)   11 1 2r r r   [You may use the result 1 ( 1)2 n r nr n    and 2 1 1 ( 1)(2 1)6 n r r n n n     ] Solution (a)   1 3 5 3 5( 1 1) 7 n r n r n n n n           (b) 2 2 2 1 1 ( )( 1) ( ) n n r n r n r n r r r nr n            2 2 22 2 1 1 1 1 ( 1) n n n n r r r n r n r r n r n               1 1(2 )(2 1)(4 1) ( 1)(2 1)6 6n n n n n n      ( 1) ( 1 2 ) ( )2 nn n n n n     [2(2 1)(4 1) ( 1)(2 1) 3( 1)(3 1) 6 ]6 n n n n n n n n          2 2 2[2(8 6 1) (2 3 1) 3(3 2 1) 6 ]6 n n n n n n n n          2(23 3 2)6 n n n   . (c) From the graphic calculator,   11 1 2 40961.5r r r   .

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ _______________________________________________________________ Additional Practice Questions for Chapter 7A: Sequences and Series Page 2 of 5 2 VJC Prelim 9740/2012/01/Q7b Given that find, in terms of a, 2 37 (2 ) a r r   , where a is an integer and a 37. [3] Solution         362 2 2 37 1 1 2 4 364 1 2 1 36 1 72 16 6 2 1 2 1 648243 a a r r r r r r a a a a a a                         3 DHS JC1 CT 9740/2014/Q3 Given that 1 ( 1)2 n r nr n    and 2 1 1 ( 1)(2 1)6 n r r n n n     , show that 3 1 3 ( 1) n r r r n n     .[3] Hence find  2 1 ( 1) 3 ( 1) n r n n r r r      , simplifying your answer. [3] Solution 2 1 1 1 3 3 33 ( 1) 3 3 ( 1)(2 1) ( 1)6 2 1 ( 1)(2 1 3)2 ( 1)( 1) (shown) n n n r r r nr r r r n n n n n n n n n n n n                          2 2 2 1 1 1 2 2 1 1 2 3 3 ( 1) 3 ( 1) ( 1) 3 ( 1) ( 2 2 1) 3 ( 1) 3 ( 1 )2 = (3 3) (2 ) 2 ( )2 n n n r n r n r n n n r r n r r r n r r r n n n r r r r n n n n n n                                       2 3 2 2 2 (3 3) (7 )2 (3 3 14 2)2 ( 11 3 2)2 n n n n n n n n n n n             2 1 ( 1)(2 1),6 n r nr n n    

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ _______________________________________________________________ Additional Practice Questions for Chapter 7A: Sequences and Series Page 3 of 5 4 CJC JC1CT 9740//2013/Q3 Using 1 ( 1)2 n r nr n    , find   100 50 ln 2 3r r    , leaving your answer in the form ln 2 ln 3A B , where A and B are constants to be found. [5] Solution      100 100 100 100 49 50 50 50 1 1 ln 2 3 ln 2 ln 3 ln 2 ln 3r r r r r r r r r                       100 49 1 1 (100 50 1)ln 2 ln 3 r r r r             100(101) 49(50)51 ln 2 ln 32 2        51 ln 2 3825 ln 3  51, 3825A B   5 VJC Promo 9758/2024/Q4 (a) The sum, nS , of the first n terms of a sequence 1 2 3, , , ...u u u is given by 1 2 375 5 n n nS    . Show that 13 5 n nu k       , where k is a constant to be determined. Find S. [4] (b) Another sequence of numbers 1 2 3, , , ...v v v is such that 1 2 2 1 n i i nv n   . Find the exact value of 11 i i v    . [3] Solution (a)   1 1 1 2 3 1 11 2 3 2 2 1 1 3 3 375 755 5 3 3 3 3 3 3 75 45 305 5 5 5 5 5 5 n n n n n n n nn n n n n n n S S u                                           21 2 3 375 75 275 5 nn n nS           As n, 23 05 n      . Hence, 75S . (b) 1 2 1 12 1 2 1 n i i nv n n     As 1, 02 1n n  . Hence, 1 1i i v    . 10 11 1 1 20 11 21 21i i i i i i v v v             .

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ _______________________________________________________________ Additional Practice Questions for Chapter 7A: Sequences and Series Page 4 of 5 6 NYJC Promo 9758/2024/Q4 During a clinical trial, the concentration U, of a specific blood agent is measured at one- hour intervals following the initial administration of the trial drug to a patient. The following readings were obtained 3 4 588, 76 and 70,U U U   where tU denotes the reading t hours after the drug was first administered. It is believed that U satisfies the relationship 1 , 0,t tU p qU t    where p and q are constants. (a) Find the values of p and q. [2] (b) Determine the initial concentration of the blood agent when the drug was initially administered. [3] (c) Describe how the concentration behaves over a long period of time. [1] Solution (a) 4 3 76 88 1U p qU p q       5 4 70 76 2U p qU p q      Solving equation (1) and (2): 32, 0.5p q  (b)    3 2 2 2 2 1 1 1 1 0 0 0 88 32 0.5 112 112 32 0.5 160 160 32 0.5 256 U p qU U U U p qU U U U p qU U U                      Alternatively,    3 2 1 0 0 0 1 1 1 1 1 12 2 2 2 2 2 1 1 12 2 2 32 32 32 32 32 32 88 32 32 32 256 U U U U U U               (c) Method 1: Method 2: As 1, , t tt U L U L    Note : This method assumes that the series is convergent. 132 2 2 64 64 L L L L L      The reading decreases and converges to 64.

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ _______________________________________________________________ Additional Practice Questions for Chapter 7A: Sequences and Series Page 5 of 5 THE END