30 f p times the pth term of an A.P. is equal to q times its qh term,

1.	30 f p times the pth term of an A.P. is equal to q times its qh term, show that the (p+g)th term of the A.P. is zero.
Answer» 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

30 f p times the pth term of an A.P. is equal to q times its qh term, show that the (p+g)th term of the A.P. is zero.

Answer»

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