Math Problem Solver

$$13,9,5,1,-3,...$$This infinite sequence is an arithmetic sequence. The n^th term rule for the sequence is $a_n=17 - 4 n$, where $n>0$. The recursive formula is $a_n=a_{n-1} - 4$, where $n>1$ and $a_{1}=13$.

$$\begin{matrix}13&&9&&5&&1&&-3\\[6pt]&-4&&-4&&-4&&-4\\[6pt]\end{matrix}$$

Explicit Formula

The formula for an arithmetic sequence where $a_1$ is the 1^st term, $d$ is the common difference, and $n$ is the term number is $$a_n=a_1 + d (n - 1)$$

Find $a_1$ and $d$: $$\begin{aligned} a_1&=13 \\ d&=-4 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 + d (n - 1) \\ &=13 + \left(-4\right) (n - 1) \\ &=17 - 4 n \end{aligned}$$

Recursive Formula

The formula for an arithmetic sequence where $a_{n-1}$ is the (n-1)^th term, $d$ is the common difference, and $n>1$ is $$a_n=a_{n-1} + d$$

Find $d$: $$\begin{aligned} d=-4 \end{aligned}$$

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