Question

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

Answer 1

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

It is given that the sum of first m terms is same as the sum of its first n terms.

∴Sm = Sn

⇒`m/2`[2a +(m − 1)d] = `n/2`[2a + (n − 1)d]

⇒2am +m(m − 1)d = 2an + n(n − 1)d

⇒2a(m − n) =[(n² − n)−(m² - m)]d

⇒2a(m − n) =[(m − n) − (n − m)(n + m)]d

⇒2a(m − n) = −(m − n)(−1 + m +n)d

⇒2a = −(m + n − 1)d .....(1)

Now,

Sum of first (m + n) terms

`= S_"m+n"`

`= (m+2)/2 [2a + (m + n - 1)d]`

`=(m+2)/2 [-(m+n-1)d + (m+n-1)d]`     [From 1]

`= "m+n"/2 xx 0`

= 0

Thus, the sum of first (m + n) terms of the AP is zero.