Math Problem Solver

$$7,4,1,...$$The 41^st term in this arithmetic sequence is $-113$.

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

Explicit Formula: $$a_n=10 - 3 n$$

Recursive Formula: $$a_n=a_{n-1} - 3, \text{where } a_{1}=7$$

$$\begin{matrix}7&&4&&1\\[6pt]&-3&&-3\\[6pt]\end{matrix}$$

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

The n^th term rule is:$$\begin{aligned} a_n&=a_1 + d (n - 1) \\ &=7 + \left(-3\right) (n - 1) \\ &=10 - 3 n \end{aligned}$$

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

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