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# If $m$ times the $m^{th}$ term of an AP is equal to $n$ times its $n^{th}$ term. find the $( m+n)^{th}$ term of the AP.

**Given:**If $m$ times the $m^{th}$ term of an AP is equal to $n$ times its $n^{th}$ term.

**To do:**To find the $( m+n)^{th}$ term of the AP.

**Solution:**

As given,

$n^{th}$ term of AP $=t_n=a+(n−1)d$

$m^{th}$ term of AP $=t_m=a+(m−1)d$

$\Rightarrow mt_m=nt_n$

$\Rightarrow m[a+(m−1)d]=n[a+(n−1)d]$

$\Rightarrow m[a+(m−1)d]−n[a+(n−1)d]=0$

$\Rightarrow a(m−n)+d[(m+n)(m−n)−(m−n)]=0$

$\Rightarrow (m−n)[a+d((m+n)−1)]=0$

$\Rightarrow a+[(m+n)−1]d=0$

But $t_{m+n}=a+[(m+n)−1]d$

$\therefore t_{m+n}=0$

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