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 22:
×2\footnotesize\maroonC{\times 2\,\Large\curvearrowright}×2\footnotesize\maroonC{\times 2\,\Large\curvearrowright}×2\footnotesize\maroonC{\times 2\,\Large\curvearrowright}
1,1,2,2,4,4,8,...8,...
Geometric sequence formulas give a(n)a(n), the nthn^{\text{th}} term of the sequence.
This is the explicit formula for the geometric sequence whose first term is k\blueD k and common ratio is r\maroonC r:
a(n)=krn1a(n)=\blueD k\cdot\maroonC r^{n-1}
This is the recursive formula of that sequence:
{a(1)=ka(n)=a(n1)r\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)\cdot\maroonC r \end{cases}
Want to learn more about geometric sequences? Check out this video.

Extending geometric sequences

Suppose we want to extend the sequence 54,18,6,...54,18,6,... We can see each term is ×13\maroonC{\times\dfrac{1}{3}} from the previous term:
×13\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}×13\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}
54,54,18,18,6,...6,...
So we simply multiply that ratio to find that the next term is 22:
×13\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}×13\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}×13\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}
54,54,18,18,6,6,2,...2,...
Problem 1
What is the next term in the sequence 12,2,8,\dfrac{1}{2},2,8, \ldots?
Want to try more problems like this? Check out this exercise.

Writing recursive formulas

Suppose we want to write a recursive formula for 54,18,6,...54,18,6,... We already know the common ratio is ×13\maroonC{\times\dfrac{1}{3}}. We can also see that the first term is 54\blueD{54}. Therefore, this is a recursive formula for the sequence:
{a(1)=54a(n)=a(n1)13\begin{cases}a(1) = \blueD{54} \\\\ a(n) = a(n-1)\cdot\maroonC{\dfrac{1}{3}} \end{cases}
Problem 1
Find kk and rr in this recursive formula of the sequence 12,2,8,\dfrac{1}{2},2,8, \ldots.
{a(1)=ka(n)=a(n1)r\begin{cases}a(1) = k \\\\ a(n) = a(n-1)\cdot r \end{cases}
k=k=
r=r=
Want to try more problems like this? Check out this exercise.

Writing explicit formulas

Suppose we want to write an explicit formula for 54,18,6,...54,18,6,... We already know the common ratio is ×13\maroonC{\times\dfrac{1}{3}} and the first term is 54\blueD{54}. Therefore, this is an explicit formula for the sequence:
a(n)=54(13)n1a(n)=\blueD{54}\cdot\left(\maroonC{\dfrac{1}{3}}\right)^{n-1}
Problem 1
Write an explicit formula for 12,2,8,\dfrac{1}{2},2,8, \ldots
a(n)=a(n)=
Want to try more problems like this? Check out this exercise.
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