Question

sum of 2nd and 5th terms of a geometric sequence is 13. the sum of the 3rd and 6th is -39.

determine:
a) the ratio "r" of the sequence
b) the first term "a" of the sequence

Answer 1

The sequence is or the form
a(1 + r + r^2 + ...)
The first term is a1 = a
The second term is a2 =a*r
The nth term is an = a*r^*(n-1)

Here is what you know:
a*r + a*r^4 = 13
a*r^2 + a*r^5 = -39
These two equations require that
-39 = r*13. Therefore
r = -3

Now that you know r, solve the first equation for a:
a*(-3 + 81) = 13
78 a = 13
a = 1/6

The sequence is:
1/6, -1/2, 3/2, -9/2, 27/2, -81/2 ...