Question

The sum of first 20 terms of an AP is equal to the sum of first 30 terms .Show that the sum of 50 terms of an AP is 0.

Answer 1

If d > 0, the sum is increasing
if d < 0, the sum is decreasing
So, if S20 = S30, d=0

That means that 20a=30a, so a=0

If a=0 and d=0, all the sums are zero.

Answer 2

a1 + a1+d + a1+2d ...... a1+(n-1)d
sum = n (a1 + an)/2

sum of first 20 = 20 (a1+a20)/2

sum of first 30 = 30 (a1 +a30)/2
so

10 a1 + 10 a20 = 15 a1 + 15 a30

5 a1 = 10 a20 - 15 a30

sum of first 50 = 50 (a1+a50)/2
= 25 a1 + 25 a50

a20 = a1 + 19d
a30 = a1 + 29d
a50 = a1 + 49d
but
5 a1 = 10 a20 - 15 a30
5 a1 = 10(a1+19d) - 15(a1+29d)
5 a1 = -5 a1 -245 d
10 a1 = -245 d
so
25a1 = -612.5 d
what is 25 a50?
25 (a1+49 d) = 25(-24.5d+49d) = 612.5d
SO
sum of first 50 = 50 (a1+a50)/2
= 25 a1 + 25 a50 -612.5d+612.5d = 0
enough already :)