Question

if 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) term is -(m+n).​

Answer 1

Step-by-step explanation:

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

Sm=n

⟹ m/2 {2a+(m−1)d} = n

⟹ 2am+m(m−1)d = 2n               ...(i)

and, Sn=m

⟹ n/2 {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)+{(m2−n2)−(m−n)}d = −2(m−n)

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

Now, 

Sm+n = 2m+n{2a+(m+n−1)d}

⟹ S m+n = m+n(−2) /2         [Using (iii)]

⟹ S m+n = −(m+n)

hope its helpful for you dear...