Question

If in an AP, the sum of $m$ terms is equal to $n$ and the sum of $n$ terms is equal to $m$, then prove that the sum of $(m + n)$terms is $-(m + n)$.​

Answer 1

Answer:

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

S

m

=n

⟹

2

m

{2a+(m−1)d}=n

⟹2am+m(m−1)d=2n ...(i)

and, S

n

=m

⟹

2

n

{2a+(n−1)d}

⟹2an+n(n−1)d=2m ...(ii)

Subtracting equation (ii) from equation (i), we get

2a(m−n)+{m(m−1)−n(n−1)}d=2n−2m

⟹2a(m−n)+{(m

2

−n

2

)−(m−n)}d=−2(m−n)

⟹2a+(m+n−1)d=−2 [On dividing both sides by (m−n)] ...(iii)

Now,

S

m+n

=

2

m+n

{2a+(m+n−1)d}

⟹S

m+n

=

2

m+n

(−2) [Using (iii)]

⟹S

m+n

=−(m+n)