Question

If the sum of first p yerms of an ap is the same as the sum of first q terms, prove that the sum of firast p+q terms = 0

Answer 1

Considering a general A.P. with starting term as 'a' and common difference as 'd'

Note S(n) means sum of n terms of the A.P.

S(p) = (p(2a + (p-1)d))/2

S(q) = (q(2a + (q-1)d))/2

S(p) = S(q)

(I will cancel out the two in both of their denominators in next step)

$2ap + {p}^{2}d - pd \: = \: 2aq + {q}^{2}d - qd \\ 2a(p - q) + ( {p}^{ 2} - {q}^{2} )d \: = (p - q)d$
Now cancel out the term (p-q) since we know that p is not equal to q.
$2a + (p + q)d \: = d \\ 2a + (p + q - 1)d = 0$
Now we need S(p+q)

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

But the term " 2a + (p+q-1)d " itself is zero.

Thus S(p+q) = 0