Arithmetic Sequences and Series

BETTER NEWS!

You know almost everything you need to know about this class of sequences, too!

A geometric sequence is nothing more than an exponential function with the specific domain of the natural numbers. The outputs of the function create the terms of the sequence just as for the arithmetic world.

Typical examples are:

{2, 4, 8, 16, 32} is a geometric term with six terms.
- This is the doubling formula with base 2.
- It's formula is .
- Notice that we show two equivalent ways of writing the formula.
- The last formula is the natural way to see the calculation. The other is the formulaic approach.
{2, 1, 0.5, 0.25, 0.125, 0.0625, . . .} is an infinite geometric sequence.
- This is the halving formula with base 2 It's formula is .
- Again there are equivalent ways to represent the sequence.

By analyzing as we did before we can create this neat formula:

The symbol is the n^th term with n position of the term in the sequence

The base b is now called the common ratio or ratio.

Let's find some n^th term formulas for geometric sequences.

Given the sequence find the n^th term formula and the value of the tenth term

By observation, the common ratio is the base 3. The formula is , The tenth term is 59049.
Given the sequence find the n^th term formula and the value of the seventh term.

By observation again, the common ratio is the base . There is a multiplier of 2 in each term. The formula is , The seventh term is .
Given the sequence find the n^th term formula and the value of the fifth term.

By observation again, the common ratio is the base . There is a multiplier of 2 in each term.

If you "observed" this base you are much sharper than most! Let's use the fact that we have a common ratio between terms.

When we suspect a geometric sequences, do calculations like those to the right where you divide the first term by the second, second by the third, etc. You should get the same ratio each time.

Then it might become obvious that there is a common multiple of 2 working in each term.

The formulas are , The fifth term is .

The next step is to find formulas for geometric series like we did for the arithmetic kind

The notations and such are just like arithmetic series, but remember these are exponential processes. The usual way of developing these formulas is by a slick rewriting process. Demonstrated to the right.

Expand the finite geometric series to show at least a few terms, including the last.
Multiply by the common ratio, r, on both sides.
Subtract the two series, which causes the terms to collapse to only the first and last.
Then solve for .

The sum for the finite geometric series is calculated by the formula:

This formula works for any finite geometric series. See how simple life is. There is no requirement to know anything more than the common ratio and the number of terms!

Examples:

For the finite sequence {2, 4, 8, 16, 32, . . . , 1024}, calculate the sum of the series.
- This is a finite series with a common ratio of 2.
- It has ten terms since .
- The sum of the series is .
For the sequence {2, 4, 8, 16, 32, . . . }, calculate the sum of the first ten terms of the series.
- This is an infinite series with a common ratio of 2.
- The sum of the first ten terms of the terms is
For the finite sequence {2, 1, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625} calculate the sum of the series.
- This is a finite series with a common ratio of 0.5.
- There are eight terms in the series.
- The sum of the series is .
For the sequence {2, -4, 8, -16, 32, -64, . . . }, calculate the sum of the first ten terms of the infinite series.
- This is a finite series with a common ratio of -2.
- It has ten terms since .
- The sum of the series is .
- Notice that this is an alternating geometric sequence. It really doesn't have any equivalent in the function world since we cannot use a negative base in an exponential function.

Now let's discuss what happens when we try to add all of the terms in an infinite series. We have already discussed why this is silly for arithmetic sequences. Fortunately, for the geometric series, this is not always silly.

It all deals with end behaviors of . It probably hasn't occur to you that this is a function in n, but that notation may suggest it. After all we used it to relate to sequences as function in the first place. For each n value we get a unique sum. That's pretty functional.

So when we talk about the sum of the infinite series, we are discussing the right-hand end behavior as n gets very large. We are looking for horizontal asymptotes. The discussion to the right is typical.

What it says is that for proper fractional ratios, we can calculate the value of the horizontal asymptote. We then call it the sum of the infinite series. In calculus courses we call this a limit process.

Go back to the infinite sequence {2, -4, 8, -16, 32, -64, . . . }. The sum of the infinite series is does not exist. The common ratio is greater than 1.

For the finite sequence {2, 1, 0.5, 0.25, 0.125, 0.0625, . . . }, the sum of the infinite series is .

Could you have guessed this answer? Well, maybe. Think about the old story about taking half a step to wall. The first two terms have a sum of three. The rest of the terms are the half step process and can never quite have a total of 1, but they get very close. The total sum must be very close to four. We call that asymptotic value the sum of the series.

Calculate the sum of the infinite sequence {4, 2, -1, 0.5, -0.25, 0.125, -0.0625, 0.03125, -0.015625, . . .}.

The common ratio is -0.5. So the sum is

A Common Application:

Recall from the sections on exponential functions, the we could calculate the value of an investment achieved through compounding using the formula . You might want to go back and review the sections.

Suppose you have a regular investment of $100 monthly with monthly compounding and a annual interest rate of 12%. What would be the total balance at the end of the second years? How much interest was paid on the deposits.

This process is called an annuity. When the deposit cycle and the compounding cycle are the same, it is an ordinary annuity. The sequence of balances for each monthly payments is a geometric sequence. There is a little trick in the sequence. If we check the balance at the end of year two, we have exactly 23 balances, not 24. The first balance calculations on the first of the second month and so on. The answer to the question is the sum of the 23 balances.

The calculation is very straight forward.

Using the formula above for n the number of compoundings.

The first term in the series is because it gets 23 compoundings.

The second is and so on.

The last deposit creates to the total balance.

When we look at the process, we need to reverse the sequence because, to answer the question, we need the common ratio is 1.01 and the value for .

The total balance for all deposits with interest is calculated to the right.

Notice $2897.78 is quite a bit better than the $2400 actually deposited.

The interest paid on the fund is $497.87.

Here's another warning: You cannot, repeat cannot, evaluate a sequence of this sort using continuous compounding. The result will be off significantly in cases with a small number of compoundings.

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