Question

Find the $n$th term of the arithmetic sequence $\left\{a_n\right\}$whose initial term $a_1$and common difference $d$are given. What is the fifty-first term?

$a_1=\sqrt{2};d=\sqrt{2}$

Short answer

The $n$th term is localid="1648725079020" $a_n=n\sqrt{2}$and the fifty-first term is $51\sqrt{2}$.

Step-by-step solution

Step 1. Given information

$a_1=\sqrt{2};d=\sqrt{2}$

Step 2. Calculation

The $n$thterm is -

$a_n=a_1+(n-1)d$

localid="1648724990758" $=\sqrt{2}+\left(n-1\right)\sqrt{2}$

$=\sqrt{2}+n\sqrt{2}-\sqrt{2}$

$=n\sqrt{2}$

and the fifty-first term is -

$a_{51}=\sqrt{2}+\left(51-1\right)\sqrt{2}$

$=\sqrt{2}+50\sqrt{2}$

$=51\sqrt{2}$