1. The first three terms of an infinite geometric sequence are 32, 16 and 8. (a) Write down the value of r. (1) (b) Find u 6. (2)
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1 Geometric Sequences and Series 1. The first three terms of an infinite geometric sequence are 32, 16 and 8. Write down the value of r. Find u 6. Find the sum to infinity of this sequence. (Total 5 marks) 2. Consider the infinite geometric sequence 3000, 1800, 1080, 648,. Find the common ratio. Find the 10 th term. Find the exact sum of the infinite sequence. 3. Consider the infinite geometric sequence 3, 3(0.9), 3(0.9) 2, 3(0.9) 3,. Write down the 10 th term of the sequence. Do not simplify your answer. Find the sum of the infinite sequence. (Total 5 marks) 4. The first four terms of a sequence are 18, 54, 162, 486. Use all four terms to show that this is a geometric sequence. IB Questionbank Maths SL 1
2 Find an expression for the n th term of this geometric sequence. If the n th term of the sequence is , find the value of n. 5. Consider the infinite geometric sequence 25, 5, 1, 0.2,. Find the common ratio. Find the 10 th term; an expression for the n th term. Find the sum of the infinite sequence. 6. Consider the infinite geometric series For this series, find the common ratio, giving your answer as a fraction in its simplest form. Find the fifteenth term of this series. Find the exact value of the sum of the infinite series. IB Questionbank Maths SL 2
3 7. Gwendolyn added the multiples of 3, from 3 to 3750 and found that Calculate s = s. 8. Find the sum of the infinite geometric series (Total 4 marks) IB Questionbank Maths SL 3
4 9. The first term of an infinite geometric sequence is 18, while the third term is 8. There are two possible sequences. Find the sum of each sequence. 10. Consider the geometric sequence 3, 6, 12, 24,. Write down the common ratio. Find the 15 th term. Consider the sequence x 3, x +1, 2x + 8,. When x = 5, the sequence is geometric. Write down the first three terms. Find the common ratio. Find the other value of x for which the sequence is geometric. IB Questionbank Maths SL 4
5 (d) For this value of x, find the common ratio; the sum of the infinite sequence. (Total 12 marks) In a geometric series, u 1 = and u4 = Find the value of r. Find the smallest value of n for which S n > 40. (Total 7 marks) IB Questionbank Maths SL 5
6 12. The diagrams below show the first four squares in a sequence of squares which are subdivided in half. The area of the shaded square A is 4 1. A A B Diagram 1 Diagram 2 A A B B C C Diagram 3 Diagram 4 Find the area of square B and of square C. (iii) Show that the areas of squares A, B and C are in geometric progression. Write down the common ratio of the progression. (5) Find the total area shaded in diagram 2. Find the total area shaded in the 8 th diagram of this sequence. Give your answer correct to six significant figures. The dividing and shading process illustrated is continued indefinitely. Find the total area shaded. (Total 11 marks) IB Questionbank Maths SL 6
7 13. The diagram shows a square ABCD of side 4 cm. The midpoints P, Q, R, S of the sides are joined to form a second square. A Q B P R Show that PQ = 2 2 cm. D S C Find the area of PQRS. The midpoints W, X, Y, Z of the sides of PQRS are now joined to form a third square as shown. A Q B W X P R Z Y D S C Write down the area of the third square, WXYZ. Show that the areas of ABCD, PQRS, and WXYZ form a geometric sequence. Find the common ratio of this sequence. IB Questionbank Maths SL 7
8 The process of forming smaller and smaller squares (by joining the midpoints) is continued indefinitely. Find the area of the 11 th square. Calculate the sum of the areas of all the squares. (Total 10 marks) 14. Portable telephones are first sold in the country Cellmania in During 1990, the number of units sold is 160. In 1991, the number of units sold is 240 and in 1992, the number of units sold is 360. In 1993 it was noticed that the annual sales formed a geometric sequence with first term 160, the 2nd and 3rd terms being 240 and 360 respectively. What is the common ratio of this sequence? Assume that this trend in sales continues. How many units will be sold during 2002? In what year does the number of units sold first exceed 5000? Between 1990 and 1992, the total number of units sold is 760. (d) What is the total number of units sold between 1990 and 2002? During this period, the total population of Cellmania remains approximately (e) Use this information to suggest a reason why the geometric growth in sales would not continue. (Total 11 marks) IB Questionbank Maths SL 8
9 15. $1000 is invested at the beginning of each year for 10 years. The rate of interest is fixed at 7.5% per annum. Interest is compounded annually. Calculate, giving your answers to the nearest dollar how much the first $1000 is worth at the end of the ten years; the total value of the investments at the end of the ten years. (Total 4 marks) 16. A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum. Write down an expression for the value of the investment after n full years. What will be the value of the investment at the end of five years? The value of the investment will exceed $ after n full years. Write down an inequality to represent this information. Calculate the minimum value of n. IB Questionbank Maths SL 9
10 17. A sum of $5000 is invested at a compound interest rate of 6.3% per annum. Write down an expression for the value of the investment after n full years. What will be the value of the investment at the end of five years? The value of the investment will exceed $10000 after n full years, Write down an inequality to represent this information. Calculate the minimum value of n. IB Questionbank Maths SL 10
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2 = 41( ) = 8897 (A1)
. Find the sum of the arithmetic series 7 + 7 + 7 +...+ 47. (Total 4 marks) R. 7 + 7 + 7 +... + 47 7 + (n )0 = 47 0(n ) = 400 n = 4 (A) 4 S 4 = ((7) + 40(0)) = 4(7 + 00) = 8897 (A) OR 4 S 4 = (7 + 47)
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