Question 1040863
Given:

1) {{{a[6] = (5/4)^4}}}, and 

2)  {{{S[3]/(a[3]+a[4]+a[5]) = 16/25}}}.

From #2,  {{{S[3]/(a[3]+a[4]+a[5]) = S[3]/(S[5]-S[2]) = 16/25}}}

==> {{{(a[1]*(r^3-1)/(r-1))/(a[1]*(r^5-1)/(r-1) -  a[1]*(r^2-1)/(r-1))}}}

= {{{(r^3-1)/(r^5-1-(r^2-1)) = (r^3-1)/(r^5-r^2) = (r^3-1)/(r^2*(r^3-1))}}}

={{{1/r^2 = 16/25}}}  ==> {{{r^2=25/16}}}  ==> r= 5/4, since r > 0 as stipulated.

==> {{{a[6] = a[1]*r^5 = (5/4)^4}}} 


==> {{{a[1]*(5/4)^5 = (5/4)^4}}}  ==> {{{a[1] = 4/5}}}

==> {{{S[3] = (4/5)*(((5/4)^3-1)/(5/4-1)) = highlight(61/20)}}}