If the sum of m terms of an A P is the same as the sum of n terms of - Maths - Arithmetic Progressions

Question

If the sum of m terms of an A P is the same as the sum of n
terms of the same A P, show that the sum of its (m+n) terms is
zero

Vimala · Accepted Answer

Let a be the first term and d be the common difference of the given AP.

Then,

$S_{m} = S_{n} \Rightarrow \frac{m}{2} {2 a + (m - 1) d} = \frac{n}{2} {2 a + (n - 1) d} \Rightarrow 2 a m + m (m - 1) d = 2 a n + n (n - 1) d \Rightarrow 2 a (m - n) + {(m^{2} - n^{2}) - (m - n)} d = 0 \Rightarrow (m - n) {2 a + (m + n - 1)} d = 0 \Rightarrow {2 a + (m + n - 1)} d = 0 [\sin c e m - n \neq 0] ....... (1)$

Let us now compute,

$S_{m + n} = \frac{m + n}{2} {2 a + (m + n - 1) d} \Rightarrow S_{m + n} = \frac{m + n}{2} \times 0 = 0 [f r o m e q u a t i o n (1)]$

That is the sum of (m +n) terms is zero.

If the sum of m terms of an A.P is the same as the sum of n terms of the same A.P, show that the sum of its (m+n) terms is zero.