Question

Use your own words to describe how sequences and series are related.

Short answer

Sequences are lists of numbers governed by rules; series are sums of sequences' terms.

Step-by-step solution

Understand Sequences

A sequence is an ordered list of numbers, each of which is called a term. The sequence follows a specific rule that defines how each term is related to its predecessor. For instance, the sequence 2, 4, 6, 8, ... follows the rule 'add 2 to the previous term.'

Understand Series

A series is the sum of the terms of a sequence. When we take the terms of a sequence and add them up, we obtain a series. Continuing with the previous example, the series formed by the sequence 2, 4, 6, 8, ... would be 2 + 4 + 6 + 8 + ....

Relating Sequences to Series

Sequences provide the terms we sum up to form a series. In other words, a series represents the cumulative effect of adding the sequential terms together. Without a sequence, we wouldn't have the set of numbers needed for the series.

Key concepts

Arithmetic Sequence

An arithmetic sequence is a type of sequence where the difference between consecutive terms is constant. This common difference is key in defining the sequence and is denoted by the letter "d". For example, in the sequence 3, 6, 9, 12, ..., the common difference is 3, since each term increases by 3 from the previous one.

To find any term in an arithmetic sequence, you can use the formula:

• \( a_n = a_1 + (n - 1) \cdot d \)

Here, \( a_n \) is the nth term, \( a_1 \) is the first term, and \( d \) is the common difference.
This straightforward relationship makes arithmetic sequences easy to work with. By understanding the common difference, you can quickly determine any term in the sequence without listing all preceding terms.

Geometric Sequence

Unlike arithmetic sequences which rely on addition, geometric sequences have a constant ratio between consecutive terms. This ratio is called the common ratio, denoted by "r". For example, in the sequence 2, 4, 8, 16, ..., each term is multiplied by 2 to get the next term. Thus, the common ratio is 2.

You can find any term in a geometric sequence using this formula:

• \( a_n = a_1 \cdot r^{(n-1)} \)

In this formula, \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.
Geometric sequences are vital in modeling situations where each term is a multiple of the previous one, such as in compound interest or population growth scenarios.

Sum of Series

The sum of a series refers to the total obtained by adding the terms of a sequence. Whether it's an arithmetic or a geometric sequence, the series formed can be summed up using specific formulas.

For an arithmetic series, the sum \( S_n \) of the first n terms is calculated using:

• \( S_n = \frac{n}{2} \left(a_1 + a_n\right) \)

Alternatively, if \( a_n \) is not known, use:

• \( S_n = \frac{n}{2} \left(2a_1 + (n-1)d\right) \)

This formula helps in determining the cumulative sum efficiently.

For a geometric series, the sum \( S_n \) is given by:

• \( S_n = a_1 \frac{1-r^n}{1-r} \), \( r eq 1 \)

These equations simplify the process of calculating the sum of a series, as manually adding each term can be tedious.