The sum of the first n terms of a geometric sequence is 255, and the

 

­Sum of first n terms of geometric sequence = \(S_{n}\) = \(\frac{a_{1} [1-r^n]}{1-r}\)  = 255 , and a1 = 1, r is the common ratio and n is the number of terms;
Sequence of the recriporcal of these terms will have a ratio of 1/r an therefore,
Sum of recriprocal of these terms = \(S_{1/n}\) = 1* (1-\(\frac{1}{r^n}\))/(1-\(\frac{1}{r}\))= \(\frac{255}{128}\)
Rearranging numerator and denominator,

\(S_{1/n}\) = \(\frac{1*[1-r^n]}{1-r}\)* \(\frac{1}{r^{n-1}}\) = \(\frac{255}{128­­­­­­­}\)

Subsitute value of \(\frac{1*[1-r^n]}{1-r}\) as 255 from first expression
==> 255*\(\frac{1}{r^{n-1}}\) = \(\frac{255}{128}\)
 ­­­­­==> \(r^{n-1}\) = \(2^7\)
=> r = 2 and n=8­­­