Question

If $m$ times the $m^{th}$ term of an A.P. is equal to $n$ times $n^{th}$ term, show that the $(m+n{)}^{th}$ term of the A.P. is zero.

Answer 1

Given,

nth term of AP $=t_n=a+(n-1)d$

mth term of AP $=t_m=a+(m-1)d$

$\Rightarrow m{t}_m=n{t}_n$

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

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

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

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

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

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

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