# Thread: One final geometric series question

1. ## One final geometric series question

The sum to infinity of a geometric series with first term a and common ratio r is 10. The sum to infinity of a second geometric series with first term a and common ratio 2r is 15.
a] find the value of r. [4]
b] find the sum of the first four terms of the series, giving your answer correct to two decimal places. [3]

Part b I will be okay with, just the first part is the trouble here.

I got two simultaneous equations from it:

10 - 10r = a - (1)
15 - 30r = a - (2)

Putting them together, I got 5 + 20r = a
Or, 5 (1 + 4r) = a, whichever is preferred.

So this means (if I'm right) that $\displaystyle \frac{a}{1 + 4r}$ = 5.

If this is right, where do I go from here?
If it is wrong, please could you help me?

2. Originally Posted by db5vry
The sum to infinity of a geometric series with first term a and common ratio r is 10. The sum to infinity of a second geometric series with first term a and common ratio 2r is 15.
a] find the value of r. [4]
I'm not sure where your equations came from...?

Try using the infinite-summation formula:

. . . . .$\displaystyle \sum_{i=0}^{\infty}\,ar^i =\, \frac{a}{1\, -\, r}\, =\, 10$

. . . . .$\displaystyle \sum_{i=0}^{\infty}\,a(2r)^i =\, \frac{a}{1\, -\, (2r)}\, =\, 15$

It is pretty easy to solve each of the above for "a=", so:

. . . . .$\displaystyle a\, =\, 10(1\, -\, r)\, =\, 15(1\, -\, 2r)$

Try solving the linear equation for the value of "r". You can then back-solve for the value of "a". :wink: