##### 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.The general formula of sequences: $ \displaystyle {{T}_{n}}={{n}^{2}}$

Square numbers form the (infinite) sequence: 1,4,9,16,25,36,………..

Square numbers may be used in other sequences: $ \displaystyle \frac{1}{4},\frac{1}{9},\frac{1}{{16}},\frac{1}{{25}},……$

2,8,18,32,50,…… each term is double a square number

### Cube numbers

A cube number is the product of multiplying a whole number by itself and then by itself again.The general formula of cube numbers: $ \displaystyle {{T}_{n}}={{n}^{3}}$

Cube numbers from the infinite sequence: 1,8,27,64,125,……

### Triangular numbers

Triangular numbers are made by arranging dots to form either equilateral or right-angled isosceles triangles.Both arrangements give the same number sequence.

The general formula of triangular numbers: $ \displaystyle {{T}_{n}}=\frac{1}{2}n(n+1)$

Triangular numbers form the (infinite) sequence: 1,3,6,10,15,……

**Fibonacci numbers **

Leonardo Fibonacci was an Italian mathematician who noticed that many natural patterns produced the sequence:

1, 1, 2, 3, 5, 8, 13, 21,…

These numbers are now called Fibonacci numbers. They have the term-to-term rule “add the two previous numbers to get the next term”.

The general formula for Fibonacci numbers: $l\displaystyle {{T}_{n}}={{T}_{{n-1}}}+{{T}_{{n-2}}}$

### Generating sequences from patterns

The diagram shows a pattern using matchsticks.

The table shows the number of matchsticks for the first five patterns.

Notice that the pattern number can be used as the position number, n, and that the numbers of matches from a sequence, just like those considered in the previous section.

The number added on each time is two but you could also see that this was true form the original diagrams. This means that the number of matches for pattern n is the same as the value pf the nth term of the sequence.

Use the ideas from the previous section to find the value of the `something`

Talking any term in the sequence from the table, for example the first: n = 1, so 2n = 2 x 1 = 2

But the first term is 3, so you need to add 1. So, nth term=2n+1

Which means that, if you let `p` be the number of matches in pattern n then, p=2n+1

Example 1: The diagram shows a pattern made with squares.

**a) **Construct a sequence table showing the first six patterns and the number of squares used.

**b)** Find a formula for the number of squares, s, in terms of the pattern number ‘n’.

4n is the formula If n = 1 then 4n = 4.

4+3 = 7

So s = 4n + 3

If n = 5 then 4n + 3 = 20 + 3 = 23, the rule is correct.

Notice that the number of square increases by 4 from shape to shape.This means that there will be a term “4n” in the formula.

Now if n = 1 then 4n = 4. The first term is seven, so you need to add three.

This means that s = 4n + 3.

Check if n = 5 then there should be 223 squares, which is correct.

**c)** How many squares will there be in pattern 100?

For pattern 100, n = 100 and $ \displaystyle s=4\times 100+3=403$

##### Be careful!

##### In any sequence n must be a positive integer. There are no negative ‘position’ for terms.For example, n can be 4 because it is possible to have a 4th term, but n cannot be -4 as it is not possible to have a 4th term.

Example 2: The position to term rule for a sequence is given as $ \displaystyle {{a}_{n}}=2n-1$. What are the first three terms of the sequence?

**Solution**: Substitute n = 1, n = 2 and n = 3 into the rule.

$ \displaystyle {{a}_{1}}=2(1)-1=1$

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

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

For the first term, n=1 and so on

The first three terms are 1, 3, and 5

Example 3: The number 123 is term in the sequence defined as $ \displaystyle {{a}_{n}}={{n}^{2}}+2$. Which term in the sequence is 123?

$ \displaystyle 123={{n}^{2}}+2$

$ \displaystyle 123-2={{n}^{2}}$

$ \displaystyle 121={{n}^{2}}$

$\displaystyle 11=n$

Find the value of n, when $ \displaystyle {{a}_{n}}=123$

$ \displaystyle \sqrt{{121}}=11$ and -11 nut n must be positive as there is no -11th term.

Tip! The nth term of a sequence can be written as an ,Tn or un. It doesn’t matter which one you choose all represent the same thing.