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Sum of First N Natural Numbers

If mth, nth...

Question

If $m t h, n t h, p t h$ terms of an AP are $a, b, c$ respectively, then $m (b - c) + n (c - a) + p (a - b)$ is

A

1

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B

a

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C

m

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D

0

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Solution

The correct option is D $0$
$T_{m} = A + (m - 1) d = a$
$T_{n} = A + (n - 1) d = b$
$T_{p} = A + (p - 1) d = c$
From these equations, we get,
$a - b = (m - n) d$
$b - c = (n - p) d$
and $c - a = (p - m) d$
Now,
$m (b - c) + n (c - a) + p (a - b) = m (n - p) d + n (p - m) d + p (m - n) d$
$= d (m n - m p + n p - n m + p m - p n)$
$= d \times 0 = 0$

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