1. Geometric Progression

Hi guys,
I am a little stuck on where to begin here and any help would be great.

This is the question I am faced with:

The sum of the first three terms of a geometric series is . The sum of the first six terms is . Find the first term and common ratio.
Now, the formula I have been using recently for working out the sum of a geometric series to n places is:

$\displaystyle S_{n} = \frac{a(1-r^n)}{1-r}$

But I am unsure how to use it in this context.

Thanks for any help!

2. Hi

You know

$\displaystyle S_{6} = \frac{a(1-r^6)}{1-r} = \frac{3367}{512}$

$\displaystyle S_{3} = \frac{a(1-r^3)}{1-r} = \frac{37}{8}$

Dividing gives

$\displaystyle \frac{1-r^6}{1-r^3} = \frac{91}{64}$

Using $\displaystyle R = r^3$ you will find a quadratic whose unknown is $\displaystyle R$

You will get R then r and finally a

3. Originally Posted by running-gag
Hi

You know

$\displaystyle S_{6} = \frac{a(1-r^6)}{1-r} = \frac{3367}{512}$

$\displaystyle S_{3} = \frac{a(1-r^3)}{1-r} = \frac{37}{8}$

Dividing gives

$\displaystyle \frac{1-r^6}{1-r^3} = \frac{91}{64}$

Using $\displaystyle R = r^3$ you will find a quadratic whose unknown is $\displaystyle R$

You will get R then r and finally a
There is no need to solve a quadratic since

$\displaystyle \frac{1-r^6}{1-r^3} =1+r^3= \frac{91}{64} \Rightarrow r^3=\frac{27}{64} \Rightarrow r=\frac{3}{4}$

4. Originally Posted by running-gag
Hi

You know

$\displaystyle S_{6} = \frac{a(1-r^6)}{1-r} = \frac{3367}{512}$

$\displaystyle S_{3} = \frac{a(1-r^3)}{1-r} = \frac{37}{8}$

Dividing gives

$\displaystyle \frac{1-r^6}{1-r^3} = \frac{91}{64}$

Using $\displaystyle R = r^3$ you will find a quadratic whose unknown is $\displaystyle R$

You will get R then r and finally a
Could you please explain the steps you took to divide those equations?

Thanks for you help

5. Originally Posted by luobo
There is no need to solve a quadratic since

$\displaystyle \frac{1-r^6}{1-r^3} =1+r^3= \frac{91}{64} \Rightarrow r^3=\frac{27}{64} \Rightarrow r=\frac{3}{4}$
Yes

I realized this after switching off my computer !

6. $\displaystyle \frac{a(1-r^6)}{1-r} = \frac{3367}{512}$

$\displaystyle \frac{a(1-r^3)}{1-r} = \frac{37}{8}$

Dividing is the same than multiplying the first equation by the inverse of the second one

$\displaystyle \frac{a(1-r^6)}{1-r} \: \frac{1-r}{a(1-r^3)} = \frac{3367}{512}\:\frac{8}{37}$

a and (1-r) are both at the numerator and the denominator, therefore they disappear

$\displaystyle \frac{1-r^6}{1-r^3} = \frac{91}{64}$

7. Thanks for the help guys, much appreciated