Question

16.1f the sum of the first m terms of an AP be n and the sum of its first n terms be m then show that the sum of its first terms is -(m+n)​

Answer 1

Question:

#I feel this is the correct question. I came across the same in 10th grade.

If the sum of first m terms of an AP be n and the sum of its first n terms be m then show that the sum of its first (m+n) terms is −(m+n).

Answer:

It is proved.

Step-by-step explanation:

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

Sₙ=n/2 (2a+(n-1)d)

Given:-

S m= n

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

2am+m(m−1)d=2n--------------------(1)

Sₙ = m

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

2an+n(n−1)d=2m----------------------(2)

On subtracting 2 from 1, we get,

2a(m−n)+[(m²-n²) - (m-n)]d=2(n−m)

(m−n)[2a+(m+n−1)d]=2(n−m)

2a+(m+n−1)d=−2-------------------------(3)

Sum of (m+n) terms of the given AP

S m+n = (m+n) / 2 (2a+(m+n-1)d)

= (m+n)/2 (-2)

= - (m+n)