1.

if an arithmetic progression if p times of pth term equal to q times of qth term then show that the p+q th term is equal to zero​

Answer»

Let a be the first TERM and d be the COMMON difference of the given A.P.

Given: (p times pth term)  = (q times QTH term)

p ap = q aq

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

ap + p²d - pd = aq + q²d - qd

ap - aq = - p²d + q²d - qd + pd

a (p - q ) = d ( q² - p² + p - q )

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

[ a² - b² = (a+b)(a-b)]

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

a = d ( - p - q + 1 ) ……………..(1)

( p + q )th term = a + (n - 1 ) d

here , n = (p+q)

(p + q)th = a + ( p + q  - 1 ) d ………….(2)

substituting a = d ( - p - q + 1 ) in eq. ( 2 )

(p + q)th = d (- q - p + 1 ) + ( p + q - 1 ) d

= -DP - dq + d + pd + qd - d

(p + q)th = 0

Hence, (p + q)th term of an A.P is zero.

HOPE THIS WILL HELP YOU....


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