# Geometric Progression Question with terms to infinity

I'm having a problem with the question stated below. I stumbled upon it during my revision And I was hoping one of you guys could help me solve it and better yet Understand how to go about it.

**A geometric Progression has the first term a, common ratio r and sum to infinity 6. A second geometric Progression has the first term 2a, common ratio r^2 and sum to infinity 7. What are the values of a and r

• Do you remember the definition for "geometric progression" -- or "geometric sequence" as some do call it? – Graham Kemp Jun 24 '18 at 6:21
• Yes I do remember – RodrigoArts Jun 24 '18 at 6:23
• Demonstrate.... – Graham Kemp Jun 24 '18 at 6:23
• 1,2,3..... (To infinity) – RodrigoArts Jun 24 '18 at 6:25
• @RodrigoArts A geometric progression is of form $a,ar,ar^2,...$ your example is an arithmetic progression – user428700 Jun 24 '18 at 6:26

$$\begin{cases} \frac{a}{1-r}=6\\ \frac{2a}{1-r^2}=7 \end{cases}$$
According to my calculations, the result is $r=\frac{5}{7}$ and $a=\frac{12}{7}$.
$$\sum_{k=0}^\infty \alpha\cdotp \rho^k= \dfrac{\alpha}{1-\rho}\qquad\text{if }\lvert \rho\rvert<1~\vee~\alpha=0$$