Math Problem Solver

$$2 \cdot \sqrt{2},\sqrt{2},0,...$$The 8^th term in this arithmetic sequence is $- 5 \sqrt{2}$.

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

Explicit Formula: $$a_n=\sqrt{2} \cdot \left(3 - n\right)$$

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

$$\begin{matrix}2 \cdot \sqrt{2}&&\sqrt{2}&&0\\[6pt]&- \sqrt{2}&&- \sqrt{2}\\[6pt]\end{matrix}$$

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 $d$: $$\begin{aligned} a_1&=2 \sqrt{2} \\ d&=- \sqrt{2} \end{aligned}$$

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

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 $d$: $$\begin{aligned} d=- \sqrt{2} \end{aligned}$$

The n^th term rule is:$$\begin{aligned} a_n&=a_{n-1} + d \\ &=a_{n-1} + \left(- \sqrt{2}\right) \\ &=a_{n-1} - \sqrt{2} \end{aligned}$$