Core Mathematics 2 Geometric Series Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Geometric Series 1
Geometric series The sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of r 1. The general term and the sum to n terms are required. The proof of the sum formula should be known. Core Mathematics 2 Geometric Series 2
Finding and Using the Common Ratio in a Geometric Sequence A geometric sequence is a sequence generated by multiplying the previous term by the same number. The number that you multiply by is called the common ratio. Example 1. 2, 6, 18, 54 common ratio = 3 6 2 = 3 18 6 = 3 Example 2. 12, 4, 4 3 common ratio = 1/3 4 12 = 1 3 To calculate the common ratio find u 2 u 1 or u 3 u 2 Example 3. If 3, x, 9 are the first 3 terms of a geometric sequence, find the exact value of x x 3 = 9 x ( 3) x = 27 x ( x) x 2 = 27 x = 27 x = 3 3 Core Mathematics 2 Geometric Series 3
Defining a Geometric Sequence The general term for a geometric sequence is:- u n = ar n 1 where a = 1st term, r = common difference n = term number U n = term you are calculating sequence goes a, ar, ar2, ar3, ar4, etc. Example 1. The first 4 terms of a geometric sequence are 4, 12, 36, 108. Find the 13 th term. a = 4, r = 3, n = 13 Using u n = ar n 1 U 13 = 4 3 13 1 = 4 3 12 = 2125764 Example 2. a) Find the common ratio of the geometric sequence 3, 4.5, 6.75, 10.125 Common ratio = U n + 1 u n r = 4 5 3 r = 1 5 b) Find the 20 th term a = 3, r = 1 5, n = 20 Using u n = ar n 1 c) Find the nth tem U 20 = 3 1 5 20 1 = 3 1 5 19 = 6650 51346 a = 3, r = 1 5, n = n Using u n = ar n 1 U n = 3 1 5 n 1 = 3 3 2 n 1 Core Mathematics 2 Geometric Series 4
Example 3. If the 6 th term of a geometric sequence is 32 and the 3 rd term is 4. Find the first term and the common ratio. Using u n = ar n 1 u 6 = ar 6 1 u 3 = ar 3 1 u 6 = 32 u 3 = 4 ar 5 = 32 ar 2 = 4 By comparing the ratios ar 5 ar = 32 2 4 r 3 = 8 r = 2 Substitute the value of r into one of your equations r = 2 ar 2 = 4 a 2 2 = 4 4a = 4 a = 1 So the common ratio is 2 and the first term is 1 Core Mathematics 2 Geometric Series 5
Growth and Decay Problems Remember for increases the multiplier is 1.?, if it s a decrease the multiplier is 0.? Example 1. If property is increasing by 7% per year, what is the multiplier? 100% + 7% = 107% 107% = 107 100 = 1 07 The multiplier = 1 07 Example 2. What is the first term in the geometric progression 3, 6, 12, 24 which will exceed one million? a = 3, r = 2 u n > 1000000 n = n using ar n 1 > 1000000 3 2 n 1 > 1000000 2 n 1 > 1000000 3 To solve unknown powers we use logs log 2 n 1 > log 1000000 3 (n 1) log 2 > log 1000000 3 n 1 > log333333 33 log 2 n 1 > 18 35 (2dp) n > 19 35 n = 20 The 20th term is the first term to exceed 1 million Core Mathematics 2 Geometric Series 6
Finding the Partial Sum of a Geometric Series General formula for a partial sum is:- S n = a(rn 1) r 1 where r > 1 Why? or S n = a(1 rn ) 1 r where r < 1 If S n = a + ar + ar 2 + ar 3 + ar 4 +...ar n 1 ( r) r S n = ar + ar 2 + ar 3 + ar 4 + ar 5 +...ar n (subtract equations) rs n S n = a + ar n or S n rs n = a ar n S n (r 1) = a(r n 1) S n (1 r) = a(1 r n ) S n = a(rn 1) r 1 S n = a(1 rn ) 1 r Example 1. Find the sum of the first 9 terms of the geometric series 18+12+8+ a = 18 r = 12 18 n = 9 using S = a(r n 1) n r 1 S 9 = 18(0 66669 1) 0 6666 1 S 9 = 18(0 93988) 0 333333 S 9 = 53 9945 S n = 54 (2sf) Example 2. Find the sum of the following geometric series 0.5 + 1 + 2 + 4..+1024 Core Mathematics 2 Geometric Series 7
a = 0 5 r = 2 u n = 1024 n = n using u n = ar n 1 1024 = 0 5 2 n 1 log 2048 = log 2 n 1 log 2048 = (n 1) log 2 log 2048 = n 1 log 2 11 = n 1 12 = n First we need to know how many terms we are adding together. This answer means we want to add together the first 12 terms. a = 0 5 r = 2 n = 12 using S n = a(rn 1) r 1 S 12 = 0 5(212 1) 2 1 S 12 = 0 5(4095) 1 S 12 = 2047 5 the sum of the first 12 terms is 2047.5 Example 3. Find r = 1 10 3 2 r Using this information the sequence begins 6, 12, 24 (3x2 1, 3x2 2 etc) a = 6 n = 10 r = 2 S n = a(rn 1) r 1 S 10 = 6(210 1) 2 1 S 10 = 6 1023 1 S 10 = 6138 Core Mathematics 2 Geometric Series 8
Finding the Sum to Infinity The total of all the terms in a series is called the sum to infinity. This occurs if r < 1 (which makes it a convergent geometric series and r > -1. Formula for the sum to infinity:- S = a 1 r when 1 < r < 1 Example 1. Find the sum to infinity of the sequence 49, 14, 4 a = 49 r = 2 7 S = = a 1 r 49 1 0 2857 = 68 6 Example 2. Find the first 4 terms of the geometric series if the first term is 12 and the sum to infinity is 24. a = 12 S = 24 we need to find r S = 24 = a 1 r 12 1 r 24(1 r) = 12 24 24r = 12 12 = 24r 0 5 = r Because r = 0.5 12 0 5 = 6 6 0 5 = 3 3 0 5 = 1 5 first 4 terms are 12, 6, 3, 1.5 Core Mathematics 2 Geometric Series 9
Homework Questions 1 Finding and Using the Common Ratio 1. Find the common ratio for the following sequences a) 3, 12, 48, 192 b) 6, 9, 13.5, 20.25 c) 5, 45, 405, 3645 d) 7, 8.75, 10.9375, 13.671875 e) 11, 33, 99, 297 2. If 16, x, 9 are the first 3 terms of a geometric sequence, find the exact value of x 3. If 5, y, 31.25 are the first 3 terms of a geometric sequence, find the exact value of y 4. Find the common ratio and hence the 5 th term of the following sequences a) 2, -0.6, 0.18, -0.054 b) 150, 60, 24, 9.6 Core Mathematics 2 Geometric Series 10
Homework Questions 2 The General Term for a Geometric Sequence 1. Find a formula for the nth term of these sequences a) 27, 9, 3, 1 b) 5, 15, 45, 135 c) 2, 2.5, 3.125, 3.90625 d) 5, -10, 20, -40 2. Find the first 3 terms of these geometric sequences given a) a = 4 r = 2 b) a = 5 r = 8 c) a = 7 r = 1.5 3. Find the 10 th term of the following sequences a) 2, 8, 32, 128 b) 96, 48, 24, 12 c) 3, 7.5, 18.75, 46.875 Core Mathematics 2 Geometric Series 11
(C2-7.3) Name: Homework Questions 3 Growth and Decay 1. A new car is purchased for 12999. Each year it depreciates in value and it is only worth 85% of the previous year s value. The owner decides to keep the car until its value goes below 5000. After how many years will the owner get rid of the car. 2. What is the first term in the geometric series 3, 3.6, 4.32, 5.184 to exceed 50 3. An oak tree is planted in 2001. If the tree is initially 2m tall and it is expected to grow at a rate of 15% per year. In what year will the tree reach 15m tall. Core Mathematics 2 Geometric Series 12
Homework Questions 4 Partial sums of Geometric Sequences 1. Find the sum of the first 10 terms of the following sequences a) 17, 51, 153, 459 b) 100, 40, 16, 6.4 c) 1, 5, 25, 125 d) 6561, 729, 81, 9 2. Find the sum of the following geometric sequences if you are given the first and the last term a) 4, 6.4, 10.24, 16.384,..67.108864 Core Mathematics 2 Geometric Series 13
b) 15, 30, 60, 120, 240,.61440 3. After how many terms does the sum of the sequence equal the following a) t 2 = 4 t 5 = 32 S n = 1024 b) t 3 = 45 t 6 = 1215 S n = 2657205 Core Mathematics 2 Geometric Series 14
Homework Questions 5 Sum to Infinity 1. Find the sum to infinity of these sequences a) 6, 3, 1.5, 0.75 b) 25, 20, 16, 12.8 c) 100, -20, 4, -0.8 2. Find the common ratio and hence the first 4 terms of these geometric sequences if the first term is given and also the sum to infinity if given. a) S = 6 2 t 1 = 5 3 b) S = 127 1 9 t1 = 143 c) S = 96 2 3 t 1 = 29 d) S = 40 t 1 = 60 Core Mathematics 2 Geometric Series 15
Past Examination Questions 1. The second and fourth terms of a geometric series are 7.2 and 5.832 respectively. The common ratio of the series is positive. For this series, find (a) the common ratio, (b) the first term, (c) the sum of the first 50 terms, giving your answer to 3 decimal places, (d) the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places. (C2, Jan 2005 Q6) 2. (a) A geometric series has first term a and common ratio r. Prove that the sum of the first n terms of the series is n a(1 r ). 1 r Mr King will be paid a salary of 35 000 in the year 2005. Mr King s contract promises a 4% increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence. (b) Find, to the nearest 100, Mr King s salary in the year 2008. Mr King will receive a salary each year from 2005 until he retires at the end of 2024. (c) Find, to the nearest 1000, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024. (C2, Jan 2005 Q9) Core Mathematics 2 Geometric Series 16
3. The first term of a geometric series is 120. The sum to infinity of the series is 480. (a) Show that the common ratio, r, is 4 3. (b) Find, to 2 decimal places, the difference between the 5th and 6th terms. (c) Calculate the sum of the first 7 terms. The sum of the first n terms of the series is greater than 300. (d) Calculate the smallest possible value of n. (C2, Jan 2006 Q4) 4. A geometric series has first term a and common ratio r. The second term of the series is 4 and the sum to infinity of the series is 25. (a) Show that 25r 2 25r + 4 = 0. (b) Find the two possible values of r. (c) Find the corresponding two possible values of a. (d) Show that the sum, Sn, of the first n terms of the series is given by Sn = 25(1 r n ). (1) Given that r takes the larger of its two possible values, (e) find the smallest value of n for which Sn exceeds 24. (C2, May2006 Q9) Core Mathematics 2 Geometric Series 17
5. A geometric series is a + ar + ar 2 +... (a) Prove that the sum of the first n terms of this series is given by Sn = a(1 r 1 r n ). (b) Find 10 k 1 k 100(2 ). (c) Find the sum to infinity of the geometric series 5 5 5 + + +... 6 18 54 (d) State the condition for an infinite geometric series with common ratio r to be convergent. (1) (C2, Jan 2007 Q10) 6. A trading company made a profit of 50 000 in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio r, r > 1. The model therefore predicts that in 2007 (Year 2) a profit of 50 000r will be made. (a) Write down an expression for the predicted profit in Year n. (1) The model predicts that in Year n, the profit made will exceed 200 000. (b) Show that n > log 4 log r + 1. Using the model with r = 1.09, (c) find the year in which the profit made will first exceed 200 000, (d) find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest 10 000. (C2, May 2007 Q8) Core Mathematics 2 Geometric Series 18
7. The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find (a) the common ratio, (b) the first term, (c) the sum of the first 20 terms, giving your answer to the nearest whole number. (C2, Jan 2008 Q2) 8. A geometric series has first term 5 and common ratio 5 4. Calculate (a) the 20th term of the series, to 3 decimal places, (b) the sum to infinity of the series. Given that the sum to k terms of the series is greater than 24.95, (c) show that k > log 0.002, log 0.8 (d) find the smallest possible value of k. (1) (C2, Jan 2008 Q6) 9. The first three terms of a geometric series are (k + 4), k and (2k 15) respectively, where k is a positive constant. (a) Show that k 2 7k 60 = 0. (b) Hence show that k = 12. (c) Find the common ratio of this series. (d) Find the sum to infinity of this series. (C2, Jan 2009 Q9) Core Mathematics 2 Geometric Series 19
10. The third term of a geometric sequence is 324 and the sixth term is 96. (a) Show that the common ratio of the sequence is 3 2. (b) Find the first term of the sequence. (c) Find the sum of the first 15 terms of the sequence. (d) Find the sum to infinity of the sequence. (C2, June 2009 Q5) 11. A car was purchased for 18 000 on 1st January. On 1st January each following year, the value of the car is 80% of its value on 1st January in the previous year. (a) Show that the value of the car exactly 3 years after it was purchased is 9216. (1) The value of the car falls below 1000 for the first time n years after it was purchased. (b) Find the value of n. An insurance company has a scheme to cover the cost of maintenance of the car. The cost is 200 for the first year, and for every following year the cost increases by 12% so that for the 3rd year the cost of the scheme is 250.88. (c) Find the cost of the scheme for the 5th year, giving your answer to the nearest penny. (d) Find the total cost of the insurance scheme for the first 15 years. (C2, Jan 2010 Q6) Core Mathematics 2 Geometric Series 20
12. The adult population of a town is 25 000 at the end of Year 1. A model predicts that the adult population of the town will increase by 3% each year, forming a geometric sequence. (a) Show that the predicted adult population at the end of Year 2 is 25 750. (b) Write down the common ratio of the geometric sequence. (1) (1) The model predicts that Year N will be the first year in which the adult population of the town exceeds 40 000. (c) Show that (d) Find the value of N. (N 1) log 1.03 > log 1.6 At the end of each year, each member of the adult population of the town will give 1 to a charity fund. Assuming the population model, (e) find the total amount that will be given to the charity fund for the 10 years from the end of Year 1 to the end of Year 10, giving your answer to the nearest 1000. (C2, June 2010 Q9) 13. The second and fifth terms of a geometric series are 750 and 6 respectively. Find (a) the common ratio of the series, (b) the first term of the series, (c) the sum to infinity of the series. (C2, Jan 2011 Q3) Core Mathematics 2 Geometric Series 21
14. The second and third terms of a geometric series are 192 and 144 respectively. For this series, find (a) the common ratio, (b) the first term, (c) the sum to infinity, (d) the smallest value of n for which the sum of the first n terms of the series exceeds 1000. (C2, May 2011 Q6) 15. A geometric series has first term a = 360 and common ratio r = 8 7. Giving your answers to 3 significant figures where appropriate, find (a) the 20th term of the series, (b) the sum of the first 20 terms of the series, (c) the sum to infinity of the series. 16. A geometric series is a + ar + ar 2 +... (C2, May 2012 Q1) (a) Prove that the sum of the first n terms of this series is given by Sn = a(1 r 1 r n ) The third and fifth terms of a geometric series are 5.4 and 1.944 respectively and all the terms in the series are positive. For this series find, (b) the common ratio, (c) the first term, (d) the sum to infinity. (C2, May 2012 Q9) Core Mathematics 2 Geometric Series 22
17. A company predicts a yearly profit of 120 000 in the year 2013. The company predicts that the yearly profit will rise each year by 5%. The predicted yearly profit forms a geometric sequence with common ratio 1.05. (a) Show that the predicted profit in the year 2016 is 138 915. (b) Find the first year in which the yearly predicted profit exceeds 200 000. (1) (5) (c) Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to the nearest pound. 18. The first three terms of a geometric series are (C2, Jan 2013 Q3) 18, 12 and p respectively, where p is a constant. Find (a) the value of the common ratio of the series, (b) the value of p, (c) the sum of the first 15 terms of the series, giving your answer to 3 decimal places. (1) (1) (C2, May 2013 Q1) 19. The first three terms of a geometric series are 4p, (3p + 15) and (5p + 20) respectively, where p is a positive constant. (a) Show that 11p 2 10p 225 = 0. (b) Hence show that p = 5. (c) Find the common ratio of this series. (d) Find the sum of the first ten terms of the series, giving your answer to the nearest integer. (C2, May 2013_R Q5) Core Mathematics 2 Geometric Series 23
20. The first term of a geometric series is 20 and the common ratio is 8 7. The sum to infinity of the series is S. (a) Find the value of S. The sum to N terms of the series is SN. (b) Find, to 1 decimal place, the value of S12. (c) Find the smallest value of N, for which S SN < 0.5. (C2, May 2014 Q6) 21. A geometric series has first term a, where a 0, and common ratio r. The sum to infinity of this series is 6 times the first term of the series. (a) Show that r = 5 6. Given that the fourth term of this series is 62.5, (b) find the value of a, (c) find the difference between the sum to infinity and the sum of the first 30 terms, giving your answer to 3 significant figures. (C2, May 2014_R Q2) 22. (i) All the terms of a geometric series are positive. The sum of the first two terms is 34 and the sum to infinity is 162. Find (a) the common ratio, (b) the first term. (ii) A different geometric series has a first term of 42 and a common ratio of 7 6. Find the smallest value of n for which the sum of the first n terms of the series exceeds 290. (C2, May 2015 Q5) Core Mathematics 2 Geometric Series 24