Question

If pth term of an AP is a and its qth term is b, then what is the sum of the first (p + q) terms of the AP? [4 MARKS]

Answer 1

Concept: 2 Marks
Application: 2 Marks

Let A and D be the first term and the common difference of the A.P.

We have,
$T_p=a=A+(p-1)D...\left(i\right)$

$T_q=b=A+(q-1)D...\left(ii\right)$

Subtracting (ii) from (i), we get,

$T_p-T_q=a-b=(p-q)D$

$\Rightarrow D=\frac{a-b}{p-q}....\left(iii\right)$

Substituting (iii) in (i), we get,

$A=a-(p-1)\left(\frac{a-b}{p-q}\right)$ ...... (iv)

Now, the sum of the first (p + q) terms is:

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

$=\left(\frac{p+q}{2}\right)\left(2\right(a-(p-1)\left(\frac{a-b}{p-q}\right))+(p+q-1\left)\left(\frac{a-b}{p-q}\right)\right)$
[From (iii) and (iv)]

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