If the sum of the first p terms equals the sum of the first q terms, then the sum of the first p+q terms is zero.
Given an arithmetic progression (A.P.) with first term 'a' and common difference 'd', let's consider the sum of the first 'p' terms and the sum of the first 'q' terms.
Sum of first 'p' terms (S_p) = (p/2) * [2a + (p-1)d]
Sum of first 'q' terms (S_q) = (q/2) * [2a + (q-1)d]
Since S_p = S_q, we can equate the above equations:
(p/2) * [2a + (p-1)d] = (q/2) * [2a + (q-1)d]
Expanding and rearranging:
a(p-q) + (p^2 - q^2)d/2 = 0
(p-q)(a + (p+q)d/2) = 0
Since p ≠ q, the only way this equation holds is if a + (p+q)d/2 = 0
This implies that the sum of the first (p+q) terms is zero.