DHS Sequences & Series (9758) Topical Revision

5. Sequences and Series 1 5. Sequences and Series 1 MJC/2008Promo/6 (a) The sum to infinity of a geometric progression is 9 2 and the second term of the progression is 2− . Find the common ratio. [3] (b) An arithmetic progression has n terms and a common difference d, where 0d  . Prove that the difference between the sum of the last k terms and the sum of the first k terms is ( )n k kd− . [4] 2 HCI/2010Promo/6 (a) An arithmetic progression has first term a and common difference d . The sum of the first 3 terms is equal to the sum of the next 6 terms. Find d in terms of a . [2] (b) The sum of the first n terms of a sequence is given by ( )21 n a − −− , where a is a constant. (i) Show that the sequence is a geometric progression, and state its common ratio in terms of a . [3] (ii) Find the set of values of a for which the sum to infinity of the sequence exists. [3] 3 SAJC/2010Promo/5(a) Given that 12 1! n nS n + =− , find (i)  Tn , giving your answer in a single fraction. [2] (ii) the exact value of 4 5 8 ...T T T+ + + . [2] 4 JJC/2009Promo/3 (a) In a geometric progression, the first term is 2009 and its common ratio is 5 7− . (i) Find the least value n such that 1 2009nU  , where nU denotes the nth term of the progression. [3] (ii) Find, correct to 2 decimal places, the sum of all the negative terms of the progression. [3] (b) 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5………., k…… is a sequence where the number k appears k times successively ( 1,2,3,4,...k = ). Find the 1000th term of the sequence. [4]

5. Sequences and Series 2 5 HCI/2009Promo/8 (i) Show that the areas of the shaded squares A1, A2, … , An in the nth diagram form a geometric progression. [2] (ii) Show that the total area of the shaded squares in the nth diagram Sn is 41134 n  − . [2] (iii) Let S be the total shaded area in the nth diagram as n → . Find the value of S. [2] (iv) Find the least value of n for which the difference between Sn and S is less than 1% of S. [3] 6 RVHS/2009Promo/7 Each year in June approximately 10% of the trees die out and in December, the workers plant 100 new trees. At the end of December 2000 there are 1200 trees in the plantation. (i) Find the number of living trees at the end of December 2002. [3] (ii) Consider 2001 as the first year. Show that