Sums of the first p, q, r terms of an A.P are a, b, c respectively. Prove that

Question

Philoid · Accepted Answer

Let x be the first term and d the common difference.

Sp = p/2[ 2x + (p-1)d] = a

Sq = q/2[2x + (q-1)d] = b

Sr = r/2[2x + (r-1)d] = c

2x + (p-1)d = 2a/p………..(i)

2x + (q-1)d = 2b/q………..(ii)

2x + (r-1)d = 2c/r………….(iii)

Eliminating x and d from (i), (ii) and (iii).

(p-q)d = 2a/p - 2b/q [Subtracting (i) and (ii)] ……(iv)

(q-r)d = 2a/q - 2c/r [Subtracting (iii) and (ii)] … (v)

[ Dividing (iv) and (v)]

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