Geometric Sequences & Series This chapter focuses on how to use find terms of a geometric sequence or series, find the sum of finite and infinite geometric series and apply the rules for geometric sequences and series to problems of growth and decay.

Geometric Sequences & Series CONTENTS: What is a geometric sequence? First term and common ratio Example 1 N th term of a sequence or series Example 2 Example 3 Assignment

What is a Geometric Sequence or Series? A sequence or series where we get from one term to the next by multiplying the previous term by a constant number is called a geometric sequence or series. These are examples of geometric sequences: (i)3, 9, 27, 81,...Multiply by 3 (ii)16, 8, 4, 2, 1, 0.5,...Multiply by ½ (iii)1, -2, 4, -8, 16,...Multiply by -2 Geometric Sequences & Series

First Term The first of a geometric sequence or series is usually called “a”. Common Ratio The constant number that each term is multiplied by is called the common ratio. To find the common ratio we take two terms and divide one by the previous one. Geometric Sequences & Series

Example 1: Find the common ratio of this sequence: 100, 25, 6.25, ,... Solution: To get the common ratio we divide a term of the sequences by the previous term. If we take the first two terms of the above sequence we get: 25/100 = ¼ Therefore r, the common ratio is ¼ Geometric Sequences & Series

n th term of a Geometric Sequence or Series If given the first term and the common ratio we can find any term of a geometric sequence or series by using the formula: n th term = ar n-1 Therefore, First term = aSecond term = ar Third term = ar 2 Fourth term = ar 3... Geometric Sequences & Series

Example 2: Find the 10 th term of the following geometric sequence: 2, 6, 18, 54,... Solution: Begin by writing down the values of a and r. a = 2r = 6/2 = 3 To get any term we substitute into the general formula: ar n-1 where n is the number of the term we want. 10 th term: ar 10-1 = ar 9 = 2(3) 9 = Geometric Sequences & Series

Example 3: The second term of a geometric sequence is 4 and the 4 th term is 16. Find: (a)common ratio(b)first term Solution: In this question we are trying to find two unknown variables, r and a. If we have two unknown variables we need to have two equations to solve. Equation 1: 2 nd term = 4 This gives:ar = 4 continued on next slide Geometric Sequences & Series

Solution continued: Equation 2:4 th term = 16 This gives:ar 3 = 16 Because our equations involve terms multiplied together we need to divide the equations to eliminate one of the variables. Divide (2) by (1) to get: ar3/ar = 16/4 r 2 = 4 r = 2 continued on next slide Geometric Sequences & Series

Solution continued: We can find the value of a by substituting the value of r into either equation. Substituting into Equation 1 gives: a(2) = 4 a = 2 Geometric Sequences & Series

Assignment This weeks assignment is a Moodle Activity. There are 5 questions to answer. Please be sure to include your working out for all questions. Deadline is 5:00pm on Monday 15 th March. Geometric Sequences & Series