Math Problem Solver

$$2,-4,8,...$$The 7^th term in this geometric sequence is $128$.

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

Explicit Formula: $$a_n=- \left(-2\right)^{n}$$

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

$$\begin{matrix}2&&-4&&8\\[6pt]&\times-2&&\times-2\\[6pt]\end{matrix}$$

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

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

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:$$\begin{aligned} a_n&=r a_{n-1} \\ &=\left(-2\right) a_{n-1} \\ &=- 2 a_{n-1} \end{aligned}$$