Question

If an ap p times of pth term equal to q times of qth term prove that (p+q)th trem are zero

Answer 1

let first term of an ap is a

and its common difference is d

given:

p[a+(p-1)d] = q[a+(q-1)d]

ap+d(p2-p) = aq+d(q2-q)

ap- aq = d(q2-p2+p-q)

a(p-q) = d[(q-p)(q+p) + p-q]

a(p-q) = d[-1(p-q)(q+p) + (p-q)]

a(p-q) = d(p-q) {-q-p+1}

a(p-q)/(p-q) = d(-q-p+1)

a= -d (p+q-1)

a+ d(p+q-1) = 0. ---(1)

we have to prove that (p+q) th term is 0

in equation 1 we proved it