SOLUTION: The sum to infinity of a geometric sequence is 27/2 while the sum of the first three terms is 13. Find the sum of the first 5 terms. Thank you for your help.

We have 

S =  =  +  =  +    (1)


Notice that the infinite sum in parentheses of the right-most side is S, again.   // It is the KEY IDEA #1.


Based on given info, replace in (1)  S  by   and replace the sum of the first three terms by 13. You will get

 = ,    or, which is the same

 = .    (2).


In (2),  replace S by    as it is given.  // It is the KEY IDEA #2.


You will get    = ,   which implies    = .


Hence,  q = .   Thus we just found the common ratio of the progression;  it is  q = . 


Now we are at the finish line.


Similar to (1), we have

S =  =  +  =  +    (3)


You can re-write it as


S =  + ,     (4)      // It is the KEY IDEA #3.


where   is the sum of the first 5 terms, which is under the question.

Now from (4)

 =  =  =  =  =  =  =  =  =  .