Geometric series problem

[Guest]

The sum of the first 2 terms of a geometric series is 90. The sum of the 6th and 7th term is -10/27. Find the sum of the first 7 terms.
I really have no idea where to start working this out.

 

[royhaas]

You must solve \(\displaystyle a+ar = 90, ar^5+ar^6 = -10/27\).

 

[Guest]

a+ar=90
a^5+a^6=-10/27

a+ar+ar^5+ar^6=2420/17
a(1+r+r^5+r^6)=2420/17

what else do i need to do to get to the answer?

 

[pka]

\(\displaystyle \begin{array}{l}
a + ra = 90\quad \Rightarrow \quad a = \frac{{90}}{{1 + r}} \\
ar^5 + ar^6 = \frac{{ - 10}}{{27}}\quad \Rightarrow \quad \frac{{90r^5 }}{{1 + r}} + \frac{{90r^6 }}{{1 + r}} = \frac{{ - 10}}{{27}} \\
\frac{{90r^5 }}{{1 + r}} + \frac{{90r^6 }}{{1 + r}} = 90r^5 = \frac{{ - 10}}{{27}} \\
\end{array}\)
Solve for r.