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 an AP is zero.

Explanation

Let the first term and the common difference of the A.P. be 'a' and 'd' respectively.
Given that,
`m . a_(m)=n . a_(n)`
`rArr m{a+(m-1)d}=n{a+(n-1)d}`
`rArr am+(m^(2)-m)d=an+(n^(2)-n)d`
`rArr a(m-n)+{(m^(2)-n^(2))-m+n}d=0`
`rArr a(m-n)+{(m-n)(m+n)-1(m-n)}d=0`
`rArr a(m-n)+(m-n)(m+n-1)d=0`
`rArr (m-n){a+(m+n-1)d}=0`
`rArr a+(m+n-1)d=0 (. :. m!=n)`
`rArr a_(m+n)=0`
`:. (m+n)`th term of the given A.P. is zero. `" " ` Hence Proved.