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# Geometric sequences review

Review geometric sequences and solve various problems involving them.

## Parts and formulas of geometric sequence

In geometric sequences, the ratio between consecutive terms is always the same. We call that ratio the common ratio.
For example, the common ratio of the following sequence is 2:
start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6
1, comma2, comma4, comma8, comma, point, point, point
Geometric sequence formulas give a, left parenthesis, n, right parenthesis, the n, start superscript, start text, t, h, end text, end superscript term of the sequence.
This is the explicit formula for the geometric sequence whose first term is start color #11accd, k, end color #11accd and common ratio is start color #ed5fa6, r, end color #ed5fa6:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, k, end color #11accd, dot, start color #ed5fa6, r, end color #ed5fa6, start superscript, n, minus, 1, end superscript
This is the recursive formula of that sequence:
$\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)\cdot\maroonC r \end{cases}$

## Extending geometric sequences

Suppose we want to extend the sequence 54, comma, 18, comma, 6, comma, point, point, point We can see each term is start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6 from the previous term:
start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6
54, comma18, comma6, comma, point, point, point
So we simply multiply that ratio to find that the next term is 2:
start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6
54, comma18, comma6, comma2, comma, point, point, point
Problem 1
What is the next term in the sequence start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots?

Want to try more problems like this? Check out this exercise.

## Writing recursive formulas

Suppose we want to write a recursive formula for 54, comma, 18, comma, 6, comma, point, point, point We already know the common ratio is start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6. We can also see that the first term is start color #11accd, 54, end color #11accd. Therefore, this is a recursive formula for the sequence:
$\begin{cases}a(1) = \blueD{54} \\\\ a(n) = a(n-1)\cdot\maroonC{\dfrac{1}{3}} \end{cases}$
Problem 1
Find k and r in this recursive formula of the sequence start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots.
$\begin{cases}a(1) = k \\\\ a(n) = a(n-1)\cdot r \end{cases}$
k, equals
r, equals

Want to try more problems like this? Check out this exercise.

## Writing explicit formulas

Suppose we want to write an explicit formula for 54, comma, 18, comma, 6, comma, point, point, point We already know the common ratio is start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6 and the first term is start color #11accd, 54, end color #11accd. Therefore, this is an explicit formula for the sequence:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, 54, end color #11accd, dot, left parenthesis, start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6, right parenthesis, start superscript, n, minus, 1, end superscript
Problem 1
Write an explicit formula for start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots
a, left parenthesis, n, right parenthesis, equals

Want to try more problems like this? Check out this exercise.