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 the number of living trees at the end of December in the nth year is given by 9 (1200) 1000(1 0.9 )10 n n +− . [3] What happens to the number of living trees in the plantation for large values of n? [2] 7 MJC/2015Promo/11 (a) The sum of the first n terms of a series is ( )41nn − . Obtain an expression for the nth term of the series. Hence prove that the series is arithmetic. [3] (b) A geometric series has common ratio r, and an arithmetic series has first term a and common difference d, where a and d are non-zero. The second, fifth and seventh terms of the arithmetic series are three consecutive terms of the geometric series. (i) Show that the geometric series is convergent. [4] (ii) It is given that the first term of the geometric series is the same as the first term of the arithmetic series. Let S be the sum to infinity of all the even - numbered terms of the geometric series and An be the sum of the first n terms of the arithmetic series. Given 0a  , find the least value of n such that 0nSA+ . [4] 2cm 2cm Diagram 1 Diagram 2 Diagram 3 A1 A1 A2 A1 A2 A3

5. Sequences and Series 3 8 DHS/2010Promo/8 A special robotic pen is programmed to draw a pattern consisting of squares in increasing sizes with no overlapping of lines. The squares have lengths which follow an arithmetic progression and the first three squares have lengths 4 cm, 6 cm and 8 cm respectively (see Figure (a)). (i) If the pattern in Figure (a) continues until the 30 th square is completed, calculate the total perimeter of the squares formed. [2] (ii) If the amount of ink in one pen can only draw up to a maximum of 10 000 cm, find the length of the largest complete square drawn when the ink runs out. [4] (iii) The robotic pen is reprogrammed to draw three circles as shown in Figure (b) which has the same total area as the three squares in Figure (a). The smallest circle of radius k cm has the same area as the smallest square of length 4 cm. Given that the radius of each circle follows a geometric progression, determine the common ratio R. [3] 9 PJC/2013Promo/14(a) (a) Mr Lee invests $A at the beginning of each year for n years, where n is a positive integer. Compound interest accumulates at the rate of R % annually and is calculated at the end of each year. Show that the total value of Mr Lee’s investment at the end of n years is given by $ 1001 1 1 100 nRA R    + + −         . [3] Suppose that R = 8. (i) Mr Lee requires his investment to be worth $50,000 at the end of 10 years. Find correct to the nearest dollar, the value of A required to achieve this. [2] (ii) His wife decides that they can afford to invest $1000 per year. In how many years will the value of their investment reach $50,000? [2] 8 cm 6 cm 4 cm Figure (a) Figure (b) k cm

5. Sequences and Series 4 10 SAJC/2009Promo/13 (a) The cost of Sonia’s new car was $P. She accepted an interest-free loan of $P, which she agreed to repay by monthly instalments. The first instalment was $1200. The instalments were increased by $50 per month so the second and third instalments were $1250 and $1300 respectively. Given that the loan was repaid in k instalments, and that the final instalment was $3250, (i) find the value of k and P. [3] The value of Sonia’s car at the end of the first year was $72000. After the first year, the value of the car depreciated, each month, by 2% of its value at the start of that month. (ii) Calculate, to the nearest dollar, the value of Sonia’s car at the end of the third year. [2] (iii) Given that the value of the car depreciated to less than 50% of the cost of the new car by the nth month. Find the value of n. [3] (b) Given that the terms of the sequence a1, a2, a3, …, an are in arithmetic progression and 1 3 r r a b =  , for r = 1, 2, 3, …, n, show that the sequence b1, b2, b3, …, bn is geometric. Given that 2 27b = and the common difference of the arithmetic progression is 2, find an expression for ra . [4] 11 SAJC/2015/Promo/10 (a) In a training exercise, athletes run from a starting point O to and from a series of points, 1 2 3,,P P P , … increasingly far away in a straight line. The distances between adjacent points are all 5 m. (Refer to the diagram below.) O 5 m 1P 5 m 2P 5 m 3P 5 m 4P 5 m 5P In the exercise, athletes start at O and run stage 1 from O to 1P and back to O, then stage 2 from O to 2P and back to O, and so on. Write down an expression for the distance run by an