Sequences: Patterns, nth Term, Recurrence Relations

1. Introduction to Sequences

A sequence is a list of numbers that follow a specific pattern. The numbers in a sequence are called terms.

We can represent a sequence as:

$$a_1, a_2, a_3, a_4, ...$$

Where $a_1$ is the first term, $a_2$ is the second term, and so on.

1.1 Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by $d$.

The formula for the nth term of an arithmetic sequence is:

$$a_n = a_1 + (n-1)d$$

Where:

• $a_n$ is the nth term
• $a_1$ is the first term
• $n$ is the term number
• $d$ is the common difference

1.2 Geometric Sequences

A geometric sequence is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio, denoted by $r$.

The formula for the nth term of a geometric sequence is:

$$a_n = a_1 \times r^{n-1}$$

Where:

• $a_n$ is the nth term
• $a_1$ is the first term
• $n$ is the term number
• $r$ is the common ratio

2. Finding the nth Term

The nth term of a sequence is a formula that allows us to find any term in the sequence without having to list all the preceding terms.

To find the nth term, we need to identify the pattern in the sequence and express it as a mathematical formula.

Example: Consider the sequence 2, 5, 8, 11, 14, ...

The difference between consecutive terms is: 5-2 = 3, 8-5 = 3, 11-8 = 3, 14-11 = 3.

Since the difference is constant, this is an arithmetic sequence with a common difference of 3.

The first term is $a_1 = 2$ and the common difference is $d = 3$.

Using the formula for the nth term of an arithmetic sequence:

$$a_n = 2 + (n-1) \times 3$$ $$a_n = 2 + 3n - 3$$ $$a_n = 3n - 1$$

So, the nth term of the sequence is $a_n = 3n - 1$.

3. Recurrence Relations

A recurrence relation defines a term in a sequence in terms of the previous terms. It provides a rule for generating the terms of a sequence.

A recurrence relation usually involves two or more preceding terms.

Example: Consider a sequence where the next term is obtained by multiplying the previous term by 2.

This can be represented by the recurrence relation:

$$a_{n+1} = 2 \times a_n$$

With the initial term $a_1 = 5$.

This is a geometric recurrence relation.

We can use this recurrence relation to generate the terms of the sequence:

$a_1 = 5$

$a_2 = 2 \times a_1 = 2 \times 5 = 10$

$a_3 = 2 \times a_2 = 2 \times 10 = 20$

$a_4 = 2 \times a_3 = 2 \times 20 = 40$

And so on...

Term Number (n)	Term (an)
1	5
2	10
3	20
4	40

4. Practice Problems

1. Find the 10th term of the arithmetic sequence: 3, 7, 11, 15, ...
2. Find the 5th term of the geometric sequence: 2, 6, 18, 54, ...
3. The nth term of a sequence is given by $a_n = 4n - 3$. Find the first three terms of the sequence.
4. A sequence is defined by the recurrence relation $a_{n+1} = 3a_n - 2$, with the first term $a_1 = 1$. Find the first four terms of the sequence.