Created by T. Madas GEOMETRIC SERIES Created by T. Madas Created by T. Madas Question 1 (**) 7,21,63,189,567, ... a) Find, using algebra, the value of the eighth term of the above sequence. b) Determine the sum of the first twelve terms of the sequence. u8 = 15309 , S12 =1860040 Question 2 (**) 59049,19683,6561,2187,729, ... For above sequence, … a) … calculate, using algebra, the value of the tenth term. b) … determine the sum of the first ten terms. c) … find the sum to infinity. u10 = 3 , S10 = 88572 , S∞ = 88573.5 Created by T. Madas Created by T. Madas Question 3 (**) 0.31, 0.0031, 0.000031, 0.00000031, ... a) Calculate, using algebra, the sum of the first five terms of the above sequence. b) Find, as an exact fraction, the sum to infinity of the series. S = 0.3131313131 , S = 31 5 ∞ 99 Question 4 (**) −1, − 1 , −− 11 , ,... 16807 2401 343 49 a) Determine, using algebra, the value of the tenth term of the above sequence. b) Find, to the nearest integer, the sum of the first ten terms of the above sequence. u10 = − 2401 , S10 ≈ − 2801 Created by T. Madas Created by T. Madas Question 5 (**) The first and second term of a geometric series is 90 and 15 , respectively. a) State the common ratio of the series. b) Calculate the sum to infinity of the series. C2I , r = 1 , S =108 6 ∞ Question 6 (**) The third and fourth term of a geometric progression is 144 and 108 , respectively. Find … a) … the common ratio of the progression. b) … the fifth term of the progression. c) … the sum to infinity of the progression. C2A , r = 3 , u = 81 , S =1024 4 5 ∞ Created by T. Madas Created by T. Madas Question 7 (**) Evaluate the sum 15 ()3× 2 r , r=1 showing clearly all the relevant workings. 196602 Question 8 (**+) A geometric series has common ratio 1 . 3 a) Find the first term of the series, given that the sum of its first four terms is 36 . b) Determine the sum to infinity of the series. a = 24.3 , S∞ = 36.45 Created by T. Madas Created by T. Madas Question 9 (**+) The second and the fifth term of a geometric series is 12 and 1.5 , respectively. a) Find the first term and the common ratio of the series. b) Calculate the sum to infinity of the series. a = 24 , r = 0.5 , S∞ = 48 Question 10 (**+) The fifth and the sixth term of a geometric series is 12 and −8 , respectively. a) Find the first term and the common ratio of the series. b) Calculate the sum to infinity of the series. a = 60.75 , r = − 2 , S = 36.45 3 ∞ Created by T. Madas Created by T. Madas Question 11 (**+) The common ratio of a geometric series is twice as large as its first term. a) Given that the sum to infinity of the series is 1, find the exact value of the first term of the series. b) Determine, as an exact fraction, the value of the fifth term of the series. C2H , a = 1 , u = 16 3 5 243 Question 12 (**+) The common ratio of a geometric series is 1 and the sum of its first three terms is 98 . 2 a) Find the first term of the series. b) Determine the sum to infinity of the series. a = 56 , u∞ =112 Created by T. Madas Created by T. Madas Question 13 (**+) The second and the fourth term of a geometric series are 3 and 1.08 , respectively. Calculate the sum to infinity of the series, given that all the terms of the series are positive. S∞ =12.5 Question 14 (**+) Miss Velibright started working as an accountant in a large law firm in the year 2001 . Her starting salary was £22,000 and her contract promised that she will be receiving a pay rise of 5% every year thereafter. Miss Velibright plans to retire in 2030 . Find to the nearest £ , … a) …her salary in the year 2030 . b) … her total earnings in employment for the years 2001 to 2030 , inclusive. C2F , £90,555 , £1,461,655 Created by T. Madas Created by T. Madas Question 15 (**+) The k th term of a geometric progression is given by −k uk =15625 × 1.25 . a) Find the first three terms of the progression. b) Find the sum to infinity of the progression. c) Evaluate the sum 10 uk , k=1 giving the answer to the nearest integer. SYN-S , u1=12500, u 2 = 10000, u 3 = 8000 , S∞ = 62500 , S10 = 55789 Created by T. Madas Created by T. Madas Question 16 (**+) The third and the sixth term of a geometric series is 4 and 6.912 , respectively. a) Find the exact value of the first term and the common ratio of the series. b) Calculate, to three significant figures, the sum of the first ten terms of the series. C2K , a = 25 , r = 6 , S ≈ 72.1 9 5 10 Question 17 (**+) The second and the fifth term of a geometric progression is 80 and 5.12 , respectively. a) Find the value of the first term and the common ratio of the progression. b) Calculate, correct to two decimal places, the difference between the sum of the first ten terms of the progression and its sum to infinity. a = 200 , r = 2 , S− S = 0.03 5 ∞ 10 Created by T. Madas Created by T. Madas Question 18 (**+) All the terms of a geometric progression are positive. The second and the fourth term of the progression is 9.6 and 6.144 , respectively. a) Find the first term and the common ratio of the progression. b) Calculate, correct to three significant figures, the sum of the first ten terms of the progression. a =12 , r = 0.8 , S10 ≈ 53.6 Question 19 (**+) The third and the sixth term of a geometric series is 54 and 1458 , respectively. a) Find the first term and the common ratio of the series. b) Determine the sum of the first ten terms of the series. C2B , a = 6 , r = 3 , S10 =177144 Created by T. Madas Created by T. Madas Question 20 (**+) 6 , 12 , 24 , 48, 96 , … , U . The geometric sequence above has k terms and its last term is U . a) Given that the sum of its terms is 1 572 858 , find the value of k . b) Determine the value of U . k =18 , U = 786432 Question 21 (**+) The first few terms of a geometric sequence are given below 16,8,4,2, ... Find … a) … the sum to infinity of the series. b) … the exact value of the twentieth term, giving the answer as a power of 2 . −15 S∞ = 32 , u20 = 2 Created by T. Madas Created by T. Madas Question 22 (***) A geometric series has first term 20480 and its sum to infinity is 81920 . a) Show that the common ratio of the series is 3 . 4 b) Calculate the difference between the fifth and the sixth term of the series. c) Determine the smallest number of terms that should be added so that their total exceeds 80000 . SYN-T , 1620 , 14 Created by T. Madas Created by T. Madas Question 23 (***) The sum of the first two terms of a geometric series is 10 and the third term is 5 . a) Find the first term and the common ratio of the series. b) Determine the sum to infinity of the series. a = 20 , r = − 0.5 , S = 40 ∞ 3 Question 24 (***) Evaluate the following sum 12 2r−1 . r=1 4095 Created by T. Madas Created by T. Madas Question 25 (***) The maximum speed, in mph, that can be achieved in each of the five gears of a sports car form a geometric progression. The maximum speed obtained in first gear is 32 mph while the car can achieve a maximum speed of 162 mph in fifth gear. Find the maximum speed that can be achieved in third gear. C2E , 72 mph Question 26 (***) The sum to infinity of a geometric progression of positive terms is 270 and the sum of its first two terms is 240 . Find the first term and the common ratio of the progression. a =180 , r = 1 3 Created by T. Madas Created by T. Madas Question 27 (***) The sum to infinity of a geometric series is four times as large as its first term. The sum of its first two terms is 2240 . Find the sum of the first five terms of the series. S5 = 3500 Created by T. Madas Created by T. Madas Question 28 (***) Grandad gave Kevin £10 on his first birthday and he increased the amount by 20% on each subsequent birthday. a) Calculate the amount of money that Kevin received from his grandad on his 10 th birthday Kevin received the last birthday amount of money from his grandad on his nth birthday and on that birthday the amount he received exceeded £1000 for the first time. b) Show clearly that 2 n > + 1. log10 () 1.2 c) State the value of n . C2V , £51.60 , n = 27 Created by T. Madas Created by T. Madas Question 29 (***+) The first three terms of a geometric series are k , 6 and 5k + 8 respectively, where k is a constant. a) Show that one of the possible values of k is 2 , and find the other. b) Given that k = 2 , find the sum of the first 10 terms of the series. k = − 18 , S = 59048 5 10 Question 30 (***+) 1++++()()()() 1x 1 x2 ++ 1 x 3 ++ 1 x 4 + ..