If m times mth term of an A.P is equal to n times nth term of the A.P. Find its (m+n)th term

If m times mth term of an A.P is equal to n times nth term of the A.P.

1.	If m times mth term of an A.P is equal to n times nth term of the A.P. Find its (m+n)th term
Answer» Thge general nth\xa0term of an AP is a + (n -1)d.\xa0From the given conditions,\xa0m (a + (m-1)d) = n( a + (n-1)d)=> am + m2d - md = an + n2d - nd=> a(m-n) + (m+n)(m-n)d - (m-n)d = 0=> (m-n) ( a + (m+n-1)d ) = 0\xa0Rejecting the non-trivial case of m=n, we assume that m and n are different.=> ( a + (m + n - 1)d ) = 0\xa0The LHS of this equation denotes the (m+n)th\xa0term of the AP, which is Zero.