Problems about series.

A series is the of consecutive terms from a(n) .

An arithmetic series is the of consecutive terms from a(n) (two words).

A geometric series is the of consecutive terms from a(n) (two words).

Consider the series: First compute the sequence of partial sums:
Make a conjecture: The sum of the first terms of this series is .

In the series there are terms, so the sum is .

Noticing patterns is called inductive productive deductive reductive reasoning, which is very important in mathematical problem solving.

But in mathematics, we usually aim to know also when (the conditions under which) and why patterns work, which requires inductive productive deductive reductive reasoning to reach careful, logical conclusions.

Here is picture that can help explain why the sum of the first odd numbers works out so nicely:

The red lines organize the dots into groups. Moving from the top left toward the bottom right, the number of dots in these groups are , and so on. More generally, if is the first group, then the group has dots, which is the odd number.

Furthermore, through the odd number, the dots are organized so that they form a square of side length , for a total of dots.

This compelling example illustrates an important result: The sequence of partial sums of a(n) series is a function of the number of terms summed.

The following examples show that patterns are sometimes much harder to see.

Consider the arithmetic series: First compute the sequence of partial sums:
Even suspecting that this sequence is a quadratic function, the pattern is hard to see.

For arithmetic series, an important and useful technique is as follows. Call the series , and then write out the series forward and backward, with the terms aligned as shown: To the right of the equals sign, the terms in each column sum to , and there are terms. (Be careful: It is easy to be “off by one.”)

So on the right, these equations sum to .

But on the left, the equations sum to . So the sum of the original series is .

Let’s generalize this approach to find the sum of the first terms of the arithmetic series that begins .

First, we need a formula for the term in the sequence .

If we call 5 the first term, then the term is .

For ease of algebra, let’s write as . Then proceed as above: The terms in each column sum to , and there are terms, so on the right, these equations sum to .

But on the left, the equations sum to . So the sum of the original series is , which is indeed a function of .

Consider a sequence whose first term is , whose second term is , and which has a constant second difference .

What type of sequence is this?

Arithmetic. Geometric. Quadratic. Something else.
Consider a sequence whose first term is , whose second term is , and which has a constant second difference .

Fill out the following table of values for this sequence. Then, find explicit formulas for both and .

We find that and that .