Sequences and Series

A sequence is an ordered list of numbers that follows a definite pattern. Its terms are written a1, a2, a3, ..., and the nth term an often has a formula, although some sequences, like the Fibonacci sequence, are defined by a recurrence relation. A series is the sum of the terms of a sequence. This chapter briefly revises arithmetic progressions, where successive terms differ by a constant, and then concentrates on geometric progressions, where each term is the previous one multiplied by a fixed non-zero common ratio r. For a GP with first term a, the nth term is ar^(n-1) and the sum of n terms is a(r^n - 1)/(r - 1) for r not equal to 1. When -1 < r < 1, the sum to infinity is a/(1 - r). You will insert geometric means between two numbers and prove that the arithmetic mean of two positive numbers is never less than their geometric mean. The ideas are applied to compound interest, population growth and bouncing balls.

Detailed Worksheet: Sequences and Series Section A - Definitions (10 marks) 1. Write the first three terms of the sequence an = n(n + 2). (2 marks) 2. Find the 10th term of the GP 5, 25, 125, .... (2 marks) 3. Find the sum of the infinite GP 1 + 1/2 + 1/4 + .... (2 marks) 4. Find the AM and GM of 4 and 16. (2 marks) 5. Distinguish between a sequence and a series. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the sum of the first 5 terms of the GP 2, 6, 18, .... (3 marks) 7. Which term of the GP 2, 2sqrt(2), 4, ... is 128? (3 marks) 8. The 4th term of a GP is 54 and the 7th term is 1458. Find the GP. (3 marks) 9. Insert three numbers between 1 and 256 so that the resulting sequence is a GP. (3 marks) 10. Find the sum of the first 20 terms of the AP 3, 7, 11, .... (3 marks) Section C - Diagrams (10 marks) 11. Plot the first five terms of the GP 1, 2, 4, 8, 16 on a graph against n and describe the shape. (4 marks) 12. Draw a diagram of a ball dropped from 10 m that rebounds to 3/4 of its previous height each time, and mark the first three heights. (3 marks) 13. Draw a table showing the first five terms of the Fibonacci sequence defined by a1 = a2 = 1 and an = a(n-1) + a(n-2). (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Derive the formula for the sum of the first n terms of a GP. Hence find the sum to n terms of 0.6 + 0.06 + 0.006 + .... (5 marks) 15. Find the sum to n terms of the series 7 + 77 + 777 + .... (5 marks) 16. Prove that the AM of two positive numbers is greater than or equal to their GM. If the AM and GM of two positive numbers are 10 and 8, find the numbers. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show the formula used in each answer.