Let \(P = 5^{1/5} \cdot 25^{1/25} \cdot 125^{1/125} \cdot 625^{1/625} \dotsm\). Then $P$ can be expressed in the form \(a^{b/c},\) where a, b, and c are positive integers. Find the smallest possible value of a + b + c.
thank you in advance for the help
+23252 51/5 x 251/25 x 1251/125 x 6251/625 x ...
= 51/5 x 52/25 x 53/125 x 54/625 x ...
= 5[ 1/5 + 2/25 + 3/125 + 4/625 + ...
Looking at only the exponent:
1/5 + 2/25 + 3/125 + 4/625 + ...
= 1/5 + 2/52 + 3/53 + 4/54 + ...
= 1/5 + (1/52 + 1/52) + (1/53 + 1/53 + 1/53) + (1/54 + 1/54 + 1/54 + 1/54) + ...
= 1/5 + 1/52 + 1/53 + 1/54 + ... = (1/5) / (1 - 1/5) = 1/4
+ 1/52 + 1/53 + 1/54 + ... = (1/52 / (1 - 1/52) = 1/20
+ 1/53 + 1/54 + ... = (1/53 / (1 - 1/53) = 1/100
....
Summing: 1/4 + 1/20 + 1/100 + ... = (1/4) / (1 - 1/5) = 5/16
So the answer is: 55/16
