Geometric Sequences
Today’s Objectives • Definition of a Geometric Sequence • Formulae for the nth term and the sum of the first n terms
Geometric Progressions (GPs) • A geometric progression is a sequence where each term is a constant multiple of the preceding term. • The constant multiple is known as the common ratio and we use the letter r. It can take any real value. • E. g. 12, 6, 3, 1. 5, 0. 75, … • This is a GP with first term 12 and common ratio of 0. 5
Geometric Formula nth term • The general formula for a term in a GP is: • un = arn-1 • where a is the first term, r is the common ratio and n is the position number
Sum of a GP • Considering the sum of the first n terms of the sequence (i. e. the series), where a is the first term, r is the common ratio: • But as with the sum of an AP this is rather long so we use a similar approach to simplify this formula.
Sum of a GP • We consider the same expression multiplied by r and subtract the two:
Sum of a GP • Hence, • If we do r. Sn – Sn instead we get, • Use the first if r < 1 and use the second if r > 1
Example 1 • The 5 th term of a GP is 8, the third term is 4, and the sum of the first ten terms is positive. Find the first term, the common ratio, and the sum of the first ten terms. • See hand written solution.
Example 2 • The sum of the first n terms of a series is 3 n – 1. Show that the terms in this series are in geometric progression and find the first term, the common ratio and the sum of the second set of n terms in this series. • See hand written solution.
Example 3 • A prize find is set up with a single investment of £ 2000 to provide an annual prize of £ 150. The fund accrues interest at 5% p. a. paid yearly. If the first prize is awarded one year after the investment, find the number of years for which the full prize can be awarded. • See hand written solution.
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