Question

If P times the pth term of an a.p. is equal to Q times the qth term find P plus Q th term of the A.P.

Answer 1

let the first term be m and common difference be d of an ap. Now,p(m+(p-1)d)=q(m+(q-1)d)=m(p-q)+d(p^2-q^2)-d(p-q)=(p-q)(m+d(p+q)-d)=(p-q)(m+d(p+q-1))=0. either p=q or m+d(p+q-1)=0 p=q is not possible so p+q term is 0

Answer 2

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