Question

if in an AP the sum of first m term=n and the sum of 1st n term=m,then prove that sum of (m+n) term is-(m+n)​

Answer 1

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

=> 2a + (m - 1)d = 2n / m → (1)

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

=> 2a + (n - 1)d = 2m / n → (2)

(1) - (2) =>

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

(m - n)d = 2(n² - m²) / mn

d = -2(m + n) / mn

From (1),

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

a = (n² + (m + n)(m - 1)) / mn

Now,

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

S_(m + n) = (m + n)((2(n² + (m + n)(m - 1)) / mn) - (2(m + n - 1)(m + n) / mn)) / 2

S_(m + n) = (m + n)(((n² + (m + n)(m - 1)) - ((m + n - 1)(m + n)) / mn))

S_(m + n) = (m + n)(((n² + m² - m + mn - n - m² - 2mn - n² + m + n) / mn))

S_(m + n) = (m + n)((- mn) / mn)

S_(m + n) = - (m + n)

Hence Proved!