Question

The first term of an arithmetic progression is $61$ and the second term is $57$ The sum of first $n$ terms is $n$. Find the value of the positive integer.

Answer 1

Given,
$\begin{array}{c}a=61,a+d=57\\ d=57-61=-4\end{array}$
The sum of first $n$ terms is:
$\begin{array}{c}\frac{n}{2}\left(2\left(61\right)+\left(n-1\right)\left(-4\right)\right)=\frac{n}{2}\left(2\right)\left(61+2-2n\right)\\ =n\left(63-2n\right)\end{array}$
Given this sum$=n$
$\begin{array}{c}n\left(63-2n\right)=n\\ 63-2n=1\\ 2n=62\\ n=\end{array}$