Question

If m times the $m^{th}$ term of an AP is equal to n times the $n^{th}$ term and its (m + n)th term is x, then the value of ${249678}^x$ is

• A. $\frac{1}{2}$
• B. 1
• C. -1
• D. 0

Answer 1

The correct option is B 1

Given:
$m{a}_m=n{a}_n$
$\Rightarrow m[a+(m-1\left)d\right]=n[a+(n-1\left)d\right]$
$\Rightarrow ma+m^{2}d-md=na+n^{2}d-nd$
$\Rightarrow (m-n)a+(m^2-n^2)d-(m-n)d=0$
$\Rightarrow (m-n)[a+\left\{\right(m+n)-1\}d]=0$
$\Rightarrow a+\left\{\right(m+n)-1\}d=0$
$\Rightarrow a_{m+n}=0$
$\Rightarrow x=0$
$\Rightarrow {249678}^x={249678}^0$
$=1$
Hence, the correct answer is option (2).