Question

If the sum of first m terms of an ap is n and sum of the first n terms is m then show that sum of first (m+n)

Answer 1

By Question,

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

2n = 2am +m(m-1)d -------(1)

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

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

Eq. (1)-(2),

2am +m(m-1)d -[2an+n(n-1)d] = 2n - 2m

2a(m-n)+[(m²-n²)-1(m-n)]d = 2(n-m)

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

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

Sum of first (m+n) terms,

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

= (m+n)/2 × -2

= m-n

Proved