In an arithmetic progression the $m$ times of m^th term is equal to $n$ times the n^th term, its (m + n)^th term will be ________.

Step 1 (Formulate AP terms): In an AP, the $k$-th term is $a_k=a+(k-1)d$. We are given that $m$ times the $m$-th term equals $n$ times the $n$-th term:

Step 2 (Expand and Group terms):

Step 3 (Divide by the non-zero difference): Factoring out $(m-n)$ (since $m\ne n$):

Dividing by $(m-n)$:

Step 4 (Identify target term): The formula for the $(m+n)$-th term of the AP is:

From Step 3, this term is exactly equal to 0.

Step 5 (Conclusion): The $(m+n)$-th term is exactly 0.