Question

The sum of n terms of an A.P is written as $S_n=pn+q{n}^2$. What is the common difference d of the A.P.? [4 MARKS]

Answer 1

Formula: 1 Mark
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Application: 1 Mark
Answer: 1 Mark

Let a be the first term of the AP. We have

$S_n=\frac{n}{2}(2a+(n-1\left)d\right)$

$S_n=an+\frac{n(n-1)d}{2}$

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

$S_n=pn+q{n}^2$

$pn+q{n}^2=(a-\frac{d}{2})n+\frac{d}{2}{n}^2$

On comparing the coefficients of $n$ and $n^2$, we get

$\frac{d}{2}=q$

$\Rightarrow d=2q$