Math Problem Solver

$$15,12,9,6,...$$This infinite sequence is an arithmetic sequence. The n^th term rule for the sequence is $a_n=18 - 3 n$, where $n>0$. The recursive formula is $a_n=a_{n-1} - 3$, where $n>1$ and $a_{1}=15$.

$$\begin{matrix}15&&12&&9&&6\\[6pt]&-3&&-3&&-3\\[6pt]\end{matrix}$$

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 $d$: $$\begin{aligned} a_1&=15 \\ d&=-3 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 + d (n - 1) \\ &=15 + \left(-3\right) (n - 1) \\ &=18 - 3 n \end{aligned}$$

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:$$\begin{aligned} a_n&=a_{n-1} + d \\ &=a_{n-1} + \left(-3\right) \\ &=a_{n-1} - 3 \end{aligned}$$