12.3 Analyze Geometric Sequences and Series

Geometric sequence

A sequence in which the ratio of any term to the previous term is constant

Common ratio

The constant ratio between consecutive terms of a geometric sequence, denoted by r

Geometric series

The expression formed by adding the terms of a geometric sequence

Example 1

Identify geometric sequences.

Tell whether the sequence 1,-4,16, -64, 256,… is geometric.

To decide whether a sequence is geometric, find the ratios of consecutive terms.

	= _____ = _-4_
= ______ = _-4_	= ______ = __-4__

Each ratio is _-4_, so the sequence _is__ geometric.

Tell whether the sequence is geometric.

1. 512, 128, 64, 8,….

not geometric

RULE FOR A GEOMETRIC SEQUENCE

The nth term of a geometric sequence with first term a₁ and common ratio r is given by: a_n = a₁r^n-1

Example 2

Write a rule for the nth term

Write a rule for the nth term of the sequence 972, -324,108, -36,… Then find a_10.

Solution

The sequence is geometric with first term a₁ = __972_ and common ratio
r = _{_______} = _{_____.} So, a rule for the nth term is:

an = a1r^n^-1	Write general rule.
= __972_ ____	Substitute for a1 and r.

The 10th term is a₁₀ = ______________ = __________

Example 3

Write a rule given a term and common ratio

One term of a geometric sequence is a₃ = -18. The common ratio is r = 3. (a) Write a rule for the nth term, (b) Graph the sequence.

a. Use the general rule to find the first term.

an = a1r^n^-1	Write general rule.
__-18__ = a1(3)_3_ -1	Substitute for an, r, and r
__-2_ = a1	Solve for ar

So, a rule for the nth term is:

an = a1r^n^-1	Write general rule.
= __-2(3)^n^-1__	Substitute for a1 and r.

b. Create a table of values for the sequence. Notice that the points lie on an exponential curve.

n	1	2	3
an	__-2__	__-6__	__-18__

n	4	5
an	__-54__	__-162__

Example 4

Write a rule given two terms

Two terms of a geometric sequence are a₂ = 10 and a₇ = -320. Find a rule for the nth term.

1. Write a system of equations using a_n = a₁r^n-1 and substituting 2 for n (Equation 1) and then 7 for n (Equation 2).

a 2 = a1r^2^-1		10 = a1r	Equation 1
a 7 = a1r^2^-1		-320 = a1r^6	Equation 2
2. Solve the system.	_______ = a1		Solve Equation 1 for a1.
	-320 = _____(r^6)		Substitute for a1 in Equation 2.
	-320 = __10r^5___		Simplify.
	__-2__ = r		Solve for r.
	10 = a1(__-2__)		Substitute in Equation 1
	__-5__ = a1		Solve for a1.
3. Find a rule an. an = a1r^n^-1			Write general rule.
an = _-5(-2)^n^-1__			Substitute.

THE SUM OF A FINITE GEOMETRIC SERIES

The sum of the first n terms of a geometric series with common ratio r ¹ 1 is:

Example 5

Find the sum of a geometric series

Find the sum of the geometric series

a1 = 3(4)^1^-1 = __3_	Identify first term.
r = __4__	Identify common ratio.
	Write rule for S13.
_________= _67,108,863_	Substitute and simplify.