The sum of the first n terms in the infinite geometric sequence {1/4, 1/8, 1/16, ...} is 255/512. Find n.

The common ratio, r,  can be determined  as  the ratio of the terms

a_n+1  / a_n    so we have....  (1/8)  / (1/4)   =  4/8  =  1/2

And we have that

255/512  =  1/4  [  1 -  (1/2)^n ] /  [  1  - 1/2  ]

255/512 =   1/4   [ 1 - (1/2)^n ]  /  (1/2)

255/512  =  (1/2)  [ 1 - (1/2)^n ]     multiply both sdes by  2

255* 2 / 512   =  1 - (1/2)^n  rearrange  as

(1/2)^n  =  1  -  510/512

(1/2)^n   =  [ 512  - 510 ] / 512    =  2 / 512  =   1/256

Take the log of both sides

log (1/2)^n  =  log (1/256)     and we can write

n  =  log (1/256) / log (1/2)

n  =   8    ⇒  8 terms