Question

If p times the p^th term is equal to q times the q^th term of an A.P. , show that the (p + q)^th term is zero.

Answer 1

Let a is the first term and d is the common difference of an AP
so,
a/ c to question,
P×Tp = q× Tq

P×{ a + ( P -1)d} = q×{ a + ( q -1)d}

Pa + P(P -1)d = qa + q(q -1)d

(P-q )a = d{ q² -q -p² +p}

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

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

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

now,

T(p + q) = a + (P+q -1)d
from equation (1)
T(P +q) = 0