If $m$ times the $m^{th}$ term of an AP is equal to $n$ times its $n^{th}$ term, show that the $(m + n)^{th}$ term of the AP is zero.

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Step 1: Given: $m[a + (m - 1)d] = n[a + (n - 1)d]$
Step 2: $a(m - n) + d(m^2 - m - n^2 + n) = 0 \Rightarrow (m - n)[a + (m + n - 1)d] = 0$
Step 3: Since $m 
eq n$: $a + (m + n - 1)d = 0 \Rightarrow a_{m+n} = 0$

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