# Need help with geometric series

1. Jan 14, 2014

### sooyong94

1. The problem statement, all variables and given/known data
In a geometric series, the first term is $a$ and the last term is $l$, If the sum of all these terms is $S$, show that the common ratio of the series is
$\frac{S-a}{S-l}$

2. Relevant equations
Sum of geometric series

3. The attempt at a solution
I was thinking to use the sum of geometric series, but I do not know how to deal with the last term. But I know that the common ratio is found by dividing the last term by the preceding term, though the problem is how do I find the preceding term?

2. Jan 14, 2014

### ehild

There is the formula for the sum of geometric series in terms of the first term a, quotient q, and the number of terms, n. Also, the last term can be expressed with a, q, n. Combine these two equations to derive the desired expression.

ehild

3. Jan 14, 2014

### sooyong94

But I don't have the formula on my textbook... :/

4. Jan 15, 2014

### ehild

5. Jan 15, 2014

### sooyong94

Unfortunately I don't seem to find one that deals with last terms... :(

6. Jan 15, 2014

### ehild

The last term is the n-th term. Take n as variable.

ehild

7. Jan 15, 2014

### sooyong94

So that would look like $T_n=ar^{n-1}$?

8. Jan 15, 2014

### ehild

Well, yes, but Tn, the last term was denoted by l.

ehild

9. Jan 15, 2014

### sooyong94

That would be $l=ar^{n-1}$...
Now let $S=\frac{a(r^{n} -1)}{r-1}$... Then $S=\frac{ar^{n}-a}{r-1}$
But $l=ar^{n-1}$... Therefore $rl=ar^{n}$. Why is it so? I don't get it here... :/
Hence $S=\frac{rl-a}{r-1}$

Now all I have to do is solve for $r$?

Last edited: Jan 15, 2014
10. Jan 15, 2014

### ehild

rn=r r((n-1)). (For example, r2=r*r; r *r^2 = r *r*r =r3... )

ehild

Last edited: Jan 15, 2014
11. Jan 15, 2014

### sooyong94

Got it. Thanks! :D

12. Jan 15, 2014

### ehild

You are welcome.

ehild

13. Jan 15, 2014

### haruspex

There's an easy way to see the truth of this.
S-a is the sum of all except the first term; S-l is the sum of all except the last term.
If you take all except the last term and multiply each by the common ratio, what set of numbers will you get?

It effectively steps the set of numbers along the sequence by one position, turning it into all except the first term.
Hence (S-l)*ratio = S-a.