Question

If A.M., G.M., and H.M. of the first and last terms of the series $100,101,102,....n-1,$ ware the terms of the series itself, then the value of w is $(100<n\le 500)$ ______

Answer 1

Answer: 400

$A.M=\frac{100+n}{2},G.M=10\sqrt{n},H.M.=\frac{200n}{100+n}$
For A.M. to be integer, $n$ must be even
For G.M. to be integer, $n$ must be a perfect square
So $n=4k^2$
$HM=\frac{200n}{100+n}=\frac{200\cdot 4K^2}{100+4K^2}=\frac{200k^2}{25+k^2}$
for H.M. to be integer, $k^2$ .must be divisible by $25$ and also
$100<n\le 500$ M
$k^2=50,75,100,125$
Hence, $n=400$ the only which satisfies