Question

IN a geometric sequence the 5th term is 60 and the first five terms are 783/2. What is the sum of the first 10 terms.

Answer 1

I will assume you mean:
the sum of the first five terms is 783/2

5th term is 60---> ar^4 = 60

the sum of the first five terms is 783/2
---> a(r^5 - 1)/(r-1) = 783/2

divide the 2nd by the 1st

a(r^5 - 1)/( (r-1)ar^4) = (783/2)÷60
(r^5 - 1)/(r^5 - r^4) = 261/40
261r^5 - 261r^4 = 40r^5 - 40
221r^5 - 261r^4 +40 = 0

a quick trial and error shows, r = 1 is a solution to the last equation.
But in a geometric series, r ≠ 1 (the sum formula breaks down)

Wolfram shows another solution at r = .88225 giving a = 99.034
for those two values:
sum(10) = a(r^10 - 1)/(r-1)
I get 600.760