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

$A.M =\frac{100+n}{2}, G.M=10\sqrt{n}, H.M.=\frac{200n}{100+n}$
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