Question

The sum of $n$ terms of an arithmetic series is $S_n=2n-n^2$. Find the first term and the common difference.

• A. $a=2;d=2$
• B. $a=0;d=1$
• C. $a=1;d=-2$
• D. $a=-1;d=2$

Answer 1

Answer: (C)

$a=1;d=-2$

The sum of n terms formula is given by,
$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$ ........... (1)
Given :- $S_n=2n-n^2$ ......... (2)
Comparing equation (1) & (2), we get
$\frac{n}{2}(2a+(n-1\left)d\right)=2n-n^2$
$2an+n^{2}d-nd=4n-2n^2$
$(2a-d)n+d{n}^2=4n-2n^2$ .......... (3)
On comparing factors of $n^2$ in equation (3)
$d=-2$
On comparing factors of $n$ in equation (3), we have
$2a-d=4$
$2a=4+d$
$2a=4-2$
$2a=2$
$a=1$
Therefore, the first term, $a=1$ and the common difference, $d=-2$.