How Do You Derive the Common Ratio in a Geometric Series?

[sooyong94]

Homework Statement

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
$\frac{S-a}{S-l}$

Homework Equations

Sum of geometric series

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?

 

[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

 

[sooyong94]

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

 

[ehild]

Check out the formulas for geometric progression. http://www.math10.com/en/algebra/geometric-progression.html

ehild

 

[sooyong94]

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

 

[ehild]

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

ehild

 

[sooyong94]

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

 

[ehild]

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

ehild

 

[sooyong94]

Well, yes, but Tn=l

ehild

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 $r$?

 

[ehild]

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... :/

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

ehild

 

[sooyong94]

Got it. Thanks! :D

 

[ehild]

You are welcome.

ehild

 

[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?

Spoiler

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.