# For $\;n \in N \;n \geq 25$ , Let A.G.H be A.M ,G.M and H.M of 25 and n . what is the least value of n such that A.G.H are all natural numbers greater than 25 .

$(a)\;49\qquad(b)\;81\qquad(c)\;169\qquad(d)\;225$

Explanation : $\;A=\large\frac{25+n}{2}$
$G=\sqrt{25n}=5\;\sqrt{n}$
$H=\large\frac{50n}{25+n}$
For A to be natural , n must be odd
For G to be natural n must be a square
Now H wii be natural number
Only if n=225 (All other choices fails).