Math Problem Solver

$$-3,-10,-17,-24,...$$This infinite sequence is an arithmetic sequence. The n^th term rule for the sequence is $a_n=4 - 7 n$, where $n>0$. The recursive formula is $a_n=a_{n-1} - 7$, where $n>1$ and $a_{1}=-3$.

$$\begin{matrix}-3&&-10&&-17&&-24\\[6pt]&-7&&-7&&-7\\[6pt]\end{matrix}$$

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 $d$: $$\begin{aligned} a_1&=-3 \\ d&=-7 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 + d (n - 1) \\ &=-3 + \left(-7\right) (n - 1) \\ &=4 - 7 n \end{aligned}$$

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 $d$: $$\begin{aligned} d=-7 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_{n-1} + d \\ &=a_{n-1} + \left(-7\right) \\ &=a_{n-1} - 7 \end{aligned}$$