If A.M., G.M., and H.M. of the first and last terms of the series of `100 , 101 , 102 ,...n-1,n` are the terms of the series itself, then the value of `ni s(100

Options

A. 200
B. 300
C. 400
D. 500

Answer

C

Explanation

`A.M.=(100+n)/2,G.M.=10sqrtn,H.M.=(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=(200n)/(100+n)=(200cdot4k^(2))/(100+4k^(2))=(200k^(2))/(25+k^(2))`
for H.M. to be integer, `k^(2)`must be divisble by 25 and also `100ltnle500`
`k^(2)=50,75,100,125`
Hence, n=400 only