Math Problem Solver

$$192,96,48,24,...$$The next term in this geometric sequence is $a_{5}=12$.

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

Explicit Formula: $$a_n=384 \cdot 2^{- n}$$

Recursive Formula: $$a_n=\frac{a_{n-1}}{2}, \text{where } a_{1}=192$$

$$\begin{matrix}192&&96&&48&&24\\[6pt]&\times\frac{1}{2}&&\times\frac{1}{2}&&\times\frac{1}{2}\\[6pt]\end{matrix}$$

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 $r$: $$\begin{aligned} a_1&=192 \\ r&=\frac{1}{2} \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 r^{n-1} \\ &=192 \left(\frac{1}{2}\right)^{n-1} \\ &=384 \cdot 2^{- n} \end{aligned}$$

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 $r$: $$\begin{aligned} r=\frac{1}{2} \end{aligned}$$

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