Question

If $m$ times the $m^{th}$ term of an AP is equal to n times the $n^{th}$ term to prove that $(m+n{)}^{th}$ term=0

Answer 1

Formula:

$t_n=a+(n-1)d$

Given,

$m\times t_m=n\times t_n$

$m[a+(m-1\left)d\right]=n[a+(n-1\left)d\right]$

$am+(m^2-m)d-an-(n^2-n)d=0$

$a(m-n)+d\left[\right(m^2-n^2)-(m-n\left)\right]=0$

$(m-n)[a+d((m+n)-1\left)\right]=0$

$a+\left[\right(m+n)-1]d=0$

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