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]

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Solution

Concept: 2 Marks
Application: 2 Marks

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

We have,
$T_{p} = a = A + (p - 1) D . . . (i)$

$T_{q} = b = A + (q - 1) D . . . (i i)$

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

$T_{p} - T_{q} = a - b = (p - q) D$

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

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

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

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

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

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

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

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