Question

The sum of n terms of an A.P. is zero. Show that the sum of next m terms is -am(m+n)/n-1, where a is the first term of the A.P.

Answer 1

Sn = 0 ________(1)
we know,
Sn = n/2{2a + (n -1)d } = 0
2a + (n -1)d = 0
d = -2a/(n -1) __________(2)
sum of next m terms( S'm) = sum of m terms - sum of n terms
e.g S'm = Sm - Sn
S'm = m/2{2a + (m-1)d } -0 { form eqn (1)
= m/2{2a + (m-1)(-2a)/n-1} [ from eqn (2)
= m/2[ 2a{ 1 - (m-1)/(n-1)}]
=m/2[2a(n -m)/(n-1)]
= ma(n - m)/(n-1)

hence,
sum of next m terms = am(n-m)/(n-1)

Answer 2

ARITHMETIC PROGRESSION AT IT'S BEST