Math Problem Solver

$$7,14,28,56,...$$The 14^th term in this geometric sequence is $57344$.

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

Explicit Formula: $$a_n=7 \cdot 2^{n - 1}$$

Recursive Formula: $$a_n=2 a_{n-1}, \text{where } a_{1}=7$$

$$\begin{matrix}7&&14&&28&&56\\[6pt]&\times2&&\times2&&\times2\\[6pt]\end{matrix}$$

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 $r$: $$\begin{aligned} a_1&=7 \\ r&=2 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 r^{n-1} \\ &=7 \left(2\right)^{n-1} \\ &=7 \cdot 2^{n - 1} \end{aligned}$$

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 $r$: $$\begin{aligned} r=2 \end{aligned}$$

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