If the sum of the mth term of an AP is equal to the sum of Nth term of an - Maths - Arithmetic Progressions

Question

If the sum of the mth term of an AP is equal to the sum of Nth
term of an AP , then show that the sum of its (m+n)th terms is
zero

Ajanta Trivedi · Accepted Answer

let the first term of the AP be a, let the common difference be d,

let the sum of the m terms be p, therefore sum of the n terms will also be p

sum of m terms $S_{m} = \frac{m}{2} (2 a + (m - 1) d) = p ...... (1)$

sum of n terms $S_{n} = \frac{n}{2} (2 a + (n - 1) d) = p .... (2)$

sum of m+n terms is

$\frac{(m + n)}{2} (2 a + (m + n - 1) d) = \frac{m}{2} (2 a + (m - 1) d) + \frac{m n}{2} d + \frac{n}{2} (2 a + (n - 1) d) + \frac{m n}{2} d = p + p + m n d = 2 p + m n d ........ (3)$

now finding d from (1) and (2)

multiplying (1) by n and (2) by m, and subtracting:

$\frac{m n}{2} (2 a) + \frac{m n}{2} (m - 1) d = n p \frac{m n}{2} (2 a) + \frac{m n}{2} (n - 1) d = m p \frac{m n}{2} (m - n) d = p (n - m) m n d = - 2 p$

substitute the value in (3), we have

sum of m+n terms =2p-2p=0

If the sum of the mth term of an AP is equal to the sum of Nth term of an AP , then show that the sum of its (m+n)th terms is zero.