Math Problem Solver

$$\frac{1}{2},\frac{2}{5},\frac{3}{10},...$$The next term in this arithmetic sequence is $a_{4}=\frac{1}{5}$.

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

Explicit Formula: $$a_n=\frac{3}{5} - \frac{n}{10}$$

Recursive Formula: $$a_n=a_{n-1} - \frac{1}{10}, \text{where } a_{1}=\frac{1}{2}$$

$$\begin{matrix}\frac{1}{2}&&\frac{2}{5}&&\frac{3}{10}\\[6pt]&- \frac{1}{10}&&- \frac{1}{10}\\[6pt]\end{matrix}$$

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 $d$: $$\begin{aligned} a_1&=\frac{1}{2} \\ d&=- \frac{1}{10} \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_1 + d (n - 1) \\ &=\frac{1}{2} + \left(- \frac{1}{10}\right) (n - 1) \\ &=\frac{3}{5} - \frac{n}{10} \end{aligned}$$

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 $d$: $$\begin{aligned} d=- \frac{1}{10} \end{aligned}$$

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