Question

If $a$ and $d$ are the first term and the common difference of an AP, $S_n$ denotes the sum of $n$ terms of an AP. If $S_n=Pn+Q{n}^2$, what are $a$ and $d$ in terms of $P$ and $Q$?

• A. a = P + Q ; d = 2Q
• B. a = P - Q ; d = 2Q
• C. a = P + Q ; d = Q/2
• D. a = P + Q ; d = Q

Answer 1

The correct option is A

a = P + Q ; d = 2Q

$S_n$ = ( $\frac{n}{2}$)(2a + (n - 1)d) = na + $\frac{n(n-1)d}{2}$ = na + $\frac{n^{2}d-nd}{2}$ = n(a - $\frac{d}{2}$) + ( $\frac{d}{2}$) $n^2$

Therefore,

$P=a-\frac{d}{2}$ and $Q=\frac{d}{2}$

$\Rightarrow a=P+Q\text{and}d=2Q$