It doesn't matter where the first term of a sequence begins. Much the same it doesn't matter too much where the first term of a geometric series begins. For the sake of making Sigma notation tidy and the math as simple as possible, we usually assume a geometric series starts at term 0. Mathematicians are some of the laziest people around. They just solve math problems day after day.
The first 1 term of the series corresponds to i = 0.
ar0 + ar1 + ar2 + ...
The first 2 terms of the series correspond to i = 0 and i = 1:
ar0 + ar1 + ar2 + ...
The first 3 terms of the series correspond to i = 0, i = 1, and i = 2:
ar0 + ar1 + ar2 + ...
Continuing in this manner, the first n terms of the series correspond to the terms from i = 0 to i = (n – 1).
ar0 + ar1 + ar2 + ... + arn – 1 + arn + ...
This means the nth partial sum of the geometric series
is
Sn = ar0 + ar1 + ar2 + ... + arn – 1
or
This is the sum of n terms. The index ranges from 0 to n – 1.
Be Careful: When finding the nth partial sum of a geometric series, the index ranges from 0 to (n – 1).
The index does not get up to n.
We promised a magic formula for finite geometric series. This time, we are going to pull a lemming out of an empty reusable grocery bag. Don't worry. The magic formula works as promised.
The nth partial sum of the geometric series
is given by
where
- r is the ratio between consecutive terms,
- a is the first term (even if it's funny looking), and
- n is the number of terms.
Be Careful: n is the number of terms, not the highest exponent.
There's a good chance you'll never need to know where this formula comes from. It's more important that you know how to use the formula properly.
Why does
The nth partial sum of a series is the sum of the first n terms of that series. Think of the finite geometric series
a + ar + ar2 + ... + arn – 1
as the first n terms of an infinite geometric series, so
Sn = a + ar + ar2 + ... + arn – 1.
We have no idea who came up with this idea or how they came up with it, but if we multiply both sides of this equation by (1 – r) and rearrange a little we end up with the magic formula.
If
Sn = a + ar + ar2 + ... + arn – 2 + arn – 1
then
(1 – r)Sn = (1 – r)(a + ar + ar2 + ... + arn – 2 + arn – 1).
If we expand the right-hand side and distribute the r, a LOT of things cancel.
Putting this back together with the left-hand side of the equation,
(1 – r)Sn = a(1 – rn).
Dividing both sides by (1 – r) gives us the magic lemming formula:
If you don't feel satisfied or enlightened right now, we understand. At least the formula comes from somewhere.
Be Careful: When entering expressions like this into your calculator, make sure you get all the parentheses in the correct places. Parenthetical errors are banes of existence for both mathematicians and computer programmers.
In order to use our magic lemming formula for finite geometric series, we need to know r, a and n. The 'ran' parameters of a geometric series are simple to find, as long as we remember what they are.
- The ratio r is usually pretty easy to find. It's the ratio of two consecutive terms.
- a is the entire first term. Don't be fooled if the first term includes the ratio as a factor.
- n is the number of terms. Don't be fooled by exponents. Count how many terms there are.
Keeping these in mind, we won't be left 'ran'ning in circles.
Geometric series are the hotdogs of the series world. But hotdogs can be prepared in an number of ways: with or without relish or chili; with or with a bun; boiled, baked or grilled... Likewise, we can write the same finite geometric series several different ways:
(I) a + ar + ar2 + ... + arn – 1
(II) 
(III) 
Although these expressions all give the value of the same geometric series, practically speaking (I) and (II) are completely different from (III).
The difference is pretty easy to see. Suppose someone gives you a and r and asks you to evaluate the series when n = 10000. With expression (III) you can pull out your calculator and do it. With expressions (I) and (II) you'd be more likely to have a temper tantrum and throw the calculator in a public water fountain.
We say the expression
is in closed form.
We liked closed form expressions in math. This means evaluating the formula always takes the same amount of work no matter how big n gets.
Whether n is 10, 1000, or 10000000000000, we
- find (1 – rn),
- multiply by a, and
- divide by (1 – r).
Closed form expressions save us all time, frustration and gray hair.
The expressions
a + ar + ar2 + ... + arn – 1
and
aren't in closed form. In order to evaluate these suckers, we have to
- find a
- find ar
- find ar2
... - find arn – 1, and finally
- add up all the terms we found in the previous steps.
That's unpleasant, to say the least. More realistically, it's like sorting a truckload of green and yellow grains of sand by grain color.
As much as we love sigma notation, the closed form is often more useful.