Calculate $t_7$ when $t_1+t_6$ and $t_3+t_4+t_5$ are given in a geometric series

Question

Suppose a geometric sequence $t_n$ about which you know that $t_1 + t_6 = 488$, and that $t_3 + t_4 + t_5 = 234$. How can you find the value of $t_7$?

I keep getting stuck on the fact that $t_2$ isn't given.

Answer 1

You have $t_1(1+q^5)=488$ from the first and $t_1(q^2+q^3+q^4)=234$ from the second equation. Combining you should get $$q^2\frac{1+q+q^2}{1+q^5}=\frac{234}{488}.$$ Its numerical roots are $q=3$, $q=-0.367167 + 0.873198i$, $q=-0.367167 - 0.873198i$, $q=-0.706221$, $q=0.526026$. It seems that $q=3$ is what you need. So you can find $t_1$ from the first equation and then $t_7=t_1\cdot q^6$.