Question

If sum of first five terms of an Ap is equal to sum of first q yerms then show that sum of first p+q terms is zero

Answer 1

Given that sum of p terms of an AP = sum of first q terms of an AP.

i.e sp = sq     ------------------ (1)

We know that sum of n terms of an AP = n/2(2a + (n - 1)  * d).

Then the sum of p terms of an AP = p/2(2a + (p - 1) * d)   ----- (1)

Then the sum of 1 terms of an AP = 1/2(2a + (q - 1) * d)     ------ (2).


Substitute (2) & (3) in (1), we get

p/2(2a + (p - 1) * d) = q/2(2a + (q - 1) * d)

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

2pa + p(p-1) * d = 2qa + q(q - 1) * d

2pa + p(p-1) * d - 2qa - q(q - 1) * d

2pa - 2qa + p(p - 1) * d  - q(q - 1) * d

2pa - 2qa + (p^2 - p) * d - (q^2 - q) * d

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

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

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

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

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

Multiply both sides with p+q/2

(p+q/2) (2a + (p + q) - 1)  * d = 0.    ---- (In the form of n/2(2a + (n - 1) * d))

Therefore the sum of first (p + q) terms is zero.


Hope this helps!

Answer 2

HOPE IT HELPS YOU.......!!!✌️

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