Sequences and Series

Sequences and series might look similar, but they are not the same. Let's look at the differences.

Sequences

A sequence can be described as a set of numbers, known as terms, that all follow a rule. They are listed in a specific order, and the rule they follow is usually a mathematical pattern. Here are some examples of sequences and the rule explained;

• \((3, 9, 15, 21, 27, 33) \) Increasing by 6

• \((72, 64, 56, 48, 40, 32)\) Decreasing by 8

• \((5, 10, 20, 40, 80, 160)\) - Multiplying by 2

Sequences can be either finite like the examples above or infinite, meaning they have no end; they can be shown like this;

• \((1, 2, 3, 4, 5, 6, \dots) \)

• \((4, 7, 10, 13, 16, \dots)\)

Due to these sequences being infinite, we can use a formula to find a specific term rather than going through the whole sequence. We will look at some of these formulas later in this article.

The different types of sequences

There are two different types of common sequences;

1. Arithmetic sequences - in this sequence, the terms increase or decrease by addition or subtraction. This difference is constant and known as the common difference or \(d\) .

2. Geometric sequences - in this type of sequence, the terms increase or decrease by multiplication or division. This difference is known as the common ratio or \(r\) .

Series

A series is an addition of the terms within a sequence, for example;

• \((3, 9, 15, 21, 27, 33)\) is a sequence and its series is \(3 + 9 + 15 + 21 + 27 + 33\)

• \((72, 64, 56, 48, 40, 32)\) is a sequence and its series is \(72 + 64 + 56 + 48 + 40 + 32\)

What are the formulas used for sequences and series?

When working with sequences and series, you may be asked to find a specific term within a sequence or the sum of a series. Here are the formulas that you can use to help you find the answers:

Formulas for sequences

There is a formula for both types of sequences, arithmetic and geometric. The formula used for finding the \(n\)^th term in an arithmetic sequence is;

\[ u_n = a + (n-1)d\]

• \(u_n\) is the \(n\)^th term
• \(a\) is the first term
• \(d\) is the common difference.

Let's have a look at an example and how we would substitute it into the formula;

Find the fifteenth term of this sequence \((5, 12, 19, 26, 33, 40, \dots )\)

• \(u_n\) is the \(n\)^th term therefore \(n=15\)
• \(a\) is the first term, so \(a=5\)
• \(d\) is the common difference, so \(d = 7\)
• then \(u_{15} = 5 + (15-1)7 = 103\)

The formula used for finding the \(n\)^th term in a geometric sequence is;

\[ u_n = ar^{n-1}\]

• \(u_n\) is the \(n\)^th term
• \(a\) is the first term
• \(r\) is the common ratio.

The common ratio is the number used to multiply or divide each term.

Let's have a look at an example and how we would substitute it into the formula;

Find the \(24\)^th term of this sequence \( (6, 12, 24, 48, 96, \dots ) \)

• \(u_n\) is the \(n\)^th term therefore \(n=24\)
• \(a\) is the first term, so \(a=6\)
• \(r\) is the common ratio, so \(r = 2\)
• then \(u_{24} = 6\cdot 2^{24-1}\)

Formulas for series

The formula used for finding the sum of the first terms of an arithmetic series is;

\[ s_n = \frac{n}{2}(2a + (n-1)d) \]

• \(s_n\) is the sum of the first \(n\) terms
• \(a\) is the first term
• \(d\) is the common difference.

Let's have a look at an example and how we would substitute it into the formula;

Find the sum of the first \(35\) terms of this series \( (2, 8, 14, 20, 26, 32, \dots ) \)

• \(s_n\) is the sum of the first \(n\) terms therefore \(n=35\)
• \(a\) is the first term, so \(a=2\)
• \(d\) is the common difference, so \(d = 6\)
• then \[s_{35} = \frac{35}{2}(2\cdot 2 + (35-1)6) \]

Two different formulas can be used when finding the sum of a geometric series. The first one is easier to use when \(r <1\) and the second when \(r> 1\);

\[ s_n = \frac{a(1-r^n)}{1-r}\]

or

\[ s_n = \frac{a(r^n-1)}{r-1} \]

• \(s_n\) is the sum of the first \(n\) terms
• \(a\) is the first term
• \(r\) is the common ratio

Let's have a look at an example and how we would substitute it into the formula;

Find the sum of the first \(50\) terms of this series \( (4, 12, 36, 108, \dots ) \)

• \(s_n\) is the sum of the first \(n\) terms therefore \(n=50\)
• \(a\) is the first term, so \(a=4\)
• \(r\) is the common ratio, so \(r = 3\)
• then \[s_{50} = \frac{4(3^n-1)}{3-1} \]

Sigma notation

The Greek letter sigma can be used to identify the sum. To use this, write the limits above and below sigma to show the terms you are using. This is shown below;

\[ \sum\limits_{r=1}^6 (2r+4) \]

This shows you that you will be finding the sequence by substituting r into the equation from \(1\) to \(6\). This will then give you the figure to create your sum.

What are the applications of sequences and series?

Sequences and series can be applied in many real-life situations, and this is also known as modeling. Many examples come from money - for example if someone saves £ 10 in first month and in second month the person saves double what he had saved in previous month and so on. This way we can use geometric series to find his savings in an year.