Question

if in an AP the sum of m terms is equals to n and the sum of n terms is equal to m then prove that sum of m+n terms is -(m+n)

Answer 1

Hello dear,

Given, sum of m terms = n

⇒ Sm = n

⇒ m/2 {2a + (m-1)d} =n

⇒ 2am +m(m-1)d = 2n ....(1)

Similarly, Sn = m

⇒ 2an + n(n-1)d= 2m .... (2)

On subtracting (2) from (1) we get

⇒ Sm - Sn = 2a(m-n) + {m(m-1) - n(n-1)} d = 2n- 2m

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

⇒ 2a + (m+n-1)d = -2.... (3)

Now, Sm+n = m+n/2 {2a+(m+n-1)d}

⇒ Sm+n = m+n/2 × (-2) [using (3)]

⇒ Sm+n = - (m+n)

[Hence proved]

Hope its help you dear ,:-)