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# For an AP, if $ m $ times the mth term equals $ \mathrm{n} $ times the $ n $ th term, prove that $ (m+n) $ th term of the AP is zero. $ (m â‰ n) $.

**Given: **

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

**To do: **

We have to prove that \( (m+n) \) th term of the AP is zero.

**Solution:**

$n^{th}$ term of the AP $=t_n=a+(nâˆ’1)d$

$m^{th}$ term of the AP $=t_m=a+(mâˆ’1)d$

$(m+n)^{th}$ term of the AP $=a+[(m+n)âˆ’1]d$

According to the question,

$\Rightarrow m \times t_m=n \times t_n$

$\Rightarrow m[a+(mâˆ’1)d]=n[a+(nâˆ’1)d]$

$\Rightarrow m[a+(mâˆ’1)d]âˆ’n[a+(nâˆ’1)d]=0$

$\Rightarrow a(mâˆ’n)+d[(m+n)(mâˆ’n)âˆ’(mâˆ’n)]=0$

$\Rightarrow (mâˆ’n)[a+d((m+n)âˆ’1)]=0$

$\Rightarrow a+[(m+n)âˆ’1]d=0$

**Hence proved.**

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