Math Problem Solver

$$1,4,16,64,...$$The 10^th term in this geometric sequence is $262144$.

The nth term in this sequence is given by the formula:

Explicit Formula: $$a_n=2^{2 n - 2}$$

Recursive Formula: $$a_n=4 a_{n-1}, \text{where } a_{1}=1$$

$$\begin{matrix}1&&4&&16&&64\\[6pt]&\times4&&\times4&&\times4\\[6pt]\end{matrix}$$

Explicit Formula

The formula for a geometric sequence where $a_1$ is the 1^st term, $r$ is the common ratio, and $n$ is the term number is $$a_n=a_1 r^{n-1}$$

Find $a_1$ and $r$: $$\begin{aligned} a_1&=1 \\ r&=4 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 r^{n-1} \\ &=1 \left(4\right)^{n-1} \\ &=2^{2 n - 2} \end{aligned}$$

Recursive Formula

The formula for a geometric sequence where $a_{n-1}$ is the (n-1)^th term, $r$ is the common ratio, and $n>1$ is $$a_n=r a_{n-1}$$

Find $r$: $$\begin{aligned} r=4 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=r a_{n-1} \\ &=\left(4\right) a_{n-1} \\ &=4 a_{n-1} \end{aligned}$$