Question

The sum of the first p terms of an AP is equal to the sum of the first q terms. What is the sum of the first p + q terms? [4 MARKS]

Answer 1

Concept: 1 Mark
Application: 2 Marks

Let the first term of the AP be a, and its common difference be d. We have:

$S_p=S_q$

$\Rightarrow \frac{p}{2}(2a+(p-1\left)d\right)=\frac{q}{2}(2a+(q-1\left)d\right)$

$\Rightarrow 2ap+p(p-1)d=2aq+q(q-1)d$

$\Rightarrow 2a(p-q)+\left(\right(p^2-q^2)-(p-q\left)\right)d=0$

$\Rightarrow 2a+(p+q-1)d=0$

$S_{p+q}=\frac{(p+q)}{2}(2a+(p+q-1\left)d\right)$

$=\frac{(p+q)}{2}\times 0$

$=0$

Thus, the sum of the first p + q terms is 0.