+0

# alg 2 help

-1
227
1

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

Mar 23, 2022

#1
+1

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