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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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