Math Problem Solver

$$6,7,8,9,...$$This infinite sequence is an arithmetic sequence. The n^th term rule for the sequence is $a_n=n + 5$, where $n>0$. The recursive formula is $a_n=a_{n-1} + 1$, where $n>1$ and $a_{1}=6$.

$$\begin{matrix}6&&7&&8&&9\\[6pt]&+1&&+1&&+1\\[6pt]\end{matrix}$$

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 $d$: $$\begin{aligned} a_1&=6 \\ d&=1 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 + d (n - 1) \\ &=6 + 1 (n - 1) \\ &=n + 5 \end{aligned}$$

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 $d$: $$\begin{aligned} d=1 \end{aligned}$$

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