Question

In an A.P., $S_p=q$ and $S_q=p$, where $S_r$ denotes the sum of the first $r$ terms of an A.P. Then, the value of $S_{p+q}$ is

• A. $0$
• B. $-(p+q)$
• C. $p+q$
• D. $pq$

Answer 1

Answer: (B)

$-(p+q)$

Let the first term of the sequence be $a$ and its common difference be $d$.
Applying sum to $n$ terms formula of A.P.,

$S_p=\frac{p}{2}(2a+(p-1\left)d\right)=q$
On expanding and simplifying the above equation, we get

$2ap+p^{2}d-pd=2q$ ...(1)

Similarly, for $S_q$

$2aq+q^{2}d-qd=2p$ ...(2)

Subtracting (2) from (1), we get
$2a(p-q)+d(p^2-q^2)-d(p-q)=-2(p-q)$
Dividing both sides by $(p-q)$, we get
$2a+d(p+q)-d=-2$
$\Rightarrow 2a+(p+q-1)d=-2$ ...(3)

Sum to $p+q$ terms formula of an A.P. is
$S_{(p+q)}=\frac{p+q}{2}[2a+(p+q-1\left)d\right]$ ...(4)

Substituting (3) in (4), we get

$S_{p+q}=\frac{p+q}{2}\times (-2)=-(p+q)$

Hence, option B is the correct answer.