Geometric Sequences

Geometric Sequences

How to solve the geometric sequence problems: formula, proof, examples, and their solutions.

Formula

A geometric sequence is a sequence whose ratios of the adjacent terms are the same. an = a1*r^(n - 1). an: nth term, a1: first term, r: common ratio

A geometric sequence is a sequence
whose ratios of the adjacent terms
are the same (= r).

an = a1rn - 1

an: nth term
a1: first term
r: common ratio

Proof

Geometric Sequences: Proof of the Formula

If the common ratio is r, then

a1 = a1
a2 = a1r1
a3 = a1r2
...
an = a1rn - 1.

Example 1

For the given geometric sequence, find an. 2, 6, 18, 54, 162, ...

Find a1 and r.

a1 = 2, r = 3

a1 = 2, r = 3

an = 2⋅3n - 1

Example 2

A geometric sequence is given below. 320, 160, 80, 40, ... If ak = 5/8, find the value of k.

Find a1 and r.

a1 = 320, r = 1/2

a1 = 320, r = 1/2

an = 320⋅(1/2)n - 1

320 = 26⋅5

Prime factorization

(1/2)n - 1 = 2-n + 1

Negative exponent

ak = 5⋅2-k + 7

And it says ak = 5/8.

So ak = 5⋅2-k + 7 = 5/8.

Exponential equations

Example 3

For a geometric sequence, a2 = -6 and a5 = 48. Find an.

Write a2 and a5
by using a1 and r.

a2 = a1r = -6
a5 = a1r4 = 48

The goal is to find a1 and r.

Find r
from a1r4 / a1r = 48 / -6.

r = -2.

Put r = -2 into a1r = -6.

Then a1 = 3.

a1 = 3, r = -2

So an = 3⋅(-2)n - 1.