##### Sequences

A sequence is a collection of objects in which repetition is allowed and order matters.

### What is a sequence number?

Number sequence is a progression of an ordered list of numbers governed by a pattern or rule.

### Types of sequences

Infinite sequence is a sequence that continues indefinitely without terminating.

Finite sequence is a sequence that has an end.

The elements of a sequence are called the terms. They are labeled as $\displaystyle {{a}_{n}},{{b}_{n}},{{x}_{n}},{{y}_{n}}$ where they represent the nth term of the sequence or the general term. The index n takes values 1,2,3……

When we write the terms of a sequence we always start from the first term and then the second and so on till the nth term when the sequence is finite.

### The rules to obtain a sequence

1. The term to term rule

2. The position to term rule

1. A term to term rule allows you to find the next number on the sequence if you know the previous term (terms).

Example 1: Find the first fourth terms of a sequence where the first term is 2 and the rule is add 3.

So we have $\displaystyle {{a}_{1}}=2$ and we add 3 for every term on the following. In the same way step by step we find

$\displaystyle {{a}_{2}}=2+3=5$

$\displaystyle {{a}_{3}}=5+3=8$

$\displaystyle {{a}_{4}}=8+3=11$

And we obtain the first fourth terms of the sequence: 2,5,8,11

2. A position to term rule allows you to compute the value of any term. When we have the general rule and we want the terms of the sequence.

Example 2: The general rule of the sequence is $\displaystyle {{a}_{n}}=5n-2$.

a) Find the first three terms of the sequence.

b) Find the 50th term of the sequence.

Solution

a) $\displaystyle {{a}_{1}}=5\cdot 1-2=3$

$\displaystyle {{a}_{2}}=5\cdot 2-2=8$

$\displaystyle {{a}_{3}}=5\cdot 3-2=13$

b) $\displaystyle {{a}_{{50}}}=5\cdot 50-2=250-2=248$

A sequence is like a set, expect the fact that:

•  The terms are in order
• The same value can appear many times

Example3: $\displaystyle \left\{ {1,2,1,2,1,2,\left. 1 \right\}} \right.$ is a sequence with alternated terms 1 and 2. If it was a set we would write only $\displaystyle \left\{ {1,\left. 2 \right\}} \right.$

### The formula of the sequence

Example 4
a) Draw a diagram to show the rule that tells you how the following sequence progresses and find the nth term. 2, 6, 10, 14, 18, 22, 26,……

b) Find the nth term of the sequence.

c) Explain how you know that the number 50 is in the sequence and work out which position it is in.

d) Explain how do you know that the number 25 is not in the sequence.

Solution

a)

If n = 3 then 4n= 43 = 12  It appears that the nth  term rule should be 4n – 2.

Try for n=5,  4n-2 = 4 ⋅ 5 – 2 = 18, so the nth  is 4n-2.

Notice that 4 is added on each time, this is the common difference. This means that the coefficient of n in the nth term will be 4. This means that “4n” will form part of your nth term rule.
Now think about any  term in the sequence, for example the third( remember that the value of n gives the position in the sequence). Try 4n to see what you get when n = 3. You get an answer of 12 but you need the third term to be 10,so you must subtract 2. Your should check this. Test it using any term,say the 5th term. Substitute n = 5 into the rule.Notice that the 5th term is indeed 18.

b) nth term, n = 40.

440-2=158

To find the 40th term in the sequence you simply need to let n=40 and substitute this into the n-th term formula.

c) 4n – 2 = 50
4n – 2 = 50
4n = 52
$\displaystyle n=\frac{{52}}{4}=13$
Since this has given a whole number, 50 must be the 13-th term in the sequence.

If the number 50 is in the sequence there must be a value of n for which 4n-2=50.Rearrange the rule to make n the subject.
Divide both sides by 4.

d) 4n – 2 = 125

4n = 127

$\displaystyle n=\frac{{127}}{4}=31,75$

Since n is in the position in the sequence it must be a whole number and it is not in this case.This means that 125 cannot be a number in the sequence.

If the number 125 is in the sequence then there must be a value of n for which 4n-2=125.Rearrange to make n the subject.

Divide both sides by 4.

Example 5
We have the sequence $\displaystyle \left\{ {2,4,6,8…….\left. . \right\}} \right.$. Find:

a) The 10th term.

b) The 100th term.

Solution

To find this we need to find the general term of the sequence, so we need to find a formula that contains “ the n”, where n is a random term form the sequence. What formula may the sequence  $\displaystyle \left\{ {2,4,6,8…….\left. . \right\}} \right.$ have.

Firstly we see that the terms of our sequence grow with 2, that´s why we can suggest that the sequence might be something like “2 multiplies n“ where n is the n-th term.

We can test it by trying the first  4-th natural numbers 1,2,3,4 and we get  2n=21 = 2, 2n=22 = 4, 2n=23 = 6 and  2n=24 =8.In this case it was easy to find out the formula.

Now that we have the general formula $\displaystyle {{a}_{n}}=2n$ we can solve our questions.

a) The 10-th term is $\displaystyle {{a}_{{10}}}=2\cdot 10=20$

b) The 100-th term is $\displaystyle {{a}_{{100}}}=2\cdot 100=200$

Now, whatever term we want we can find if we have the general formula of the sequence. If we are looking for the 13-th term  then $\displaystyle {{a}_{{13}}}=2\cdot 13=26$.

#### Two Types of sequences

1. Arithmetic Sequences

2. Geometric Sequences

#### Some special sequences

Next article

### Special sequences

Square numbers
A square number is the product of multiplying a whole number by itself. Square numbers can be represented using dots arranged to make squares.

Next article

### Special sequences

Square numbers
A square number is the product of multiplying a whole number by itself. Square numbers can be represented using dots arranged to make squares.