Question

If m times the mth term of an ap is equal to n times its nth term ,then show that (m+n) th term of the ap is zero​

Answer 1

Answer:

m×am=n×an

m{a+(m−1)d}=n{a+(n−1)d}

am+m

2

d−md=an+n

2

d−nd

am−an=−m

2

d+n

2

d−nd+md

a(m−n)=d(n

2

−m

2

+m−n)

a(m−n)=d{(n−m)(n+m)+m−n}

a(m−n)=d(m−n){−1(n+m)+1}

a=d(−n−m+1)...(1)

To prove: (m+n)th term is zero

(m+n)thterm=a+(n−1)d

here, n= number of terms

=a+(m+n−1)d=0...(2)

Substituting a=d(−n−m+1) in (2)

=d(−n−m+1)+(m+n−1)d

=−dn−dm+d+md+nd−d

=0