Math Problem Solver

$$12,19,26,33,...$$The 6^th term in this arithmetic sequence is $47$.

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

Explicit Formula: $$a_n=7 n + 5$$

Recursive Formula: $$a_n=a_{n-1} + 7, \text{where } a_{1}=12$$

$$\begin{matrix}12&&19&&26&&33\\[6pt]&+7&&+7&&+7\\[6pt]\end{matrix}$$

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 $d$: $$\begin{aligned} a_1&=12 \\ d&=7 \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 + d (n - 1) \\ &=12 + 7 (n - 1) \\ &=7 n + 5 \end{aligned}$$

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

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