Question

if m times the mth term of an AP is equal to n times the nth term then show that (m+n)th term of an AP is 0​

Answer 1

Let a be the first term and d be the common difference, then

$\rm \: m a_{m}=n a_{n}$

$\[ \begin{array}{l} \rm \Rightarrow \quad m[a+(m-1) d]=n[a+(n-1) d] \\\\ \rm \Rightarrow \quad\left[\left(m^{2}-n^{2}\right)-(m-n)\right] d=-(m-n) a \\\\ \rm \Rightarrow \quad(m+n-1) d=-a \\ \\ \rm \Rightarrow \quad a+(m+n-1) d=0 \\ \\ \rm \Rightarrow \quad a_{m+n}=0 . \end{array} \]$