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If the sum of an A.P. is the same for p as for q terms, show t

Question

If the sum of an A.P. is the same for $p$ as for $q$ terms, show that its sum for $p + q$ terms is zero.

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If the sum of an A.P. is the same for

p

as for

q

terms, show that its sum for

p + q

terms is zero.

15 mins ago

Discuss this question LIVE

15 mins ago

Text solutionVerified

Sum of

p

terms of an AP with first term

a

and common difference

d

S = \frac{p}{2} (2 a + (p - 1) d)

.......

(i)

Sum of

q

terms

= S = \frac{q}{2} (2 a + (q - 1) d)

.......

(ii)

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

\Rightarrow 2 a p + (p^{2} - p) d = 2 a q + (q^{2} - q) d

\Rightarrow 2 a (p - q) + d (p^{2} - q^{2}) - d (p - q) = 0

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

CASE 1:

p - q = 0

\Rightarrow p = q

But,

p \neq = q

Case invalid

CASE 2:

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

Sum of

p + q

terms

= \frac{p + q}{2} (2 a + (p + q - 1) d)

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

Hence, sum of

p + q

terms

= 0

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Question Text	If the sum of an A.P. is the same for $p$ as for $q$ terms, show that its sum for $p + q$ terms is zero.
Answer Type	Text solution:1
Upvotes	150

If the sum of an A.P. is the same for p as for q terms, show that its sum for p+q terms is zero.

Text solutionVerified

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If the sum of an A.P. is the same for $p$ as for $q$ terms, show that its sum for $p + q$ terms is zero.