# Sequences

A sequence is a list of numbers.
A sequence has terms and values.
The terms are the positions in the sequence.
Terms are often given the letter n, to represent the position number in the list

Example

Tom, Mary, Jane, Jack

Tom is the first person in the list.
The value of n=1 is Tom
The value of n=3 is Jane
Jack is the fourth term.

Example

3,6,9,12,15……..

The value of n=1 is 3
The value of n=4 is 12
12 is the value of the fourth term.

## To find the rule of a sequence

1. Write out the sequence.
2. Write the term numbers,n, underneath.
3. Find the difference between the values.
4. If the difference between the values is a constant, multiply each term ,n, by this number.
5. Leave a gap, draw a line, copy the sequence again.
6. What constant is needed to make the values found in step 4 the same as the sequence value ?
Fill this in.
7. Write down the expression for the nth term.

Example

Write down the rule for the nth term of the sequence
5,8,11,14………

The rule is 3n + 2

Example

What is the nth term of the sequence

1,  3,   5,   7,   9………?

The rule is 2n - 1

Example

Given the sequence    1,  4,   7,   10, …..

a) Find the next three numbers  in the sequence.
b) What is the rule for the nth term of this sequence ?
c) Find the value of the 20th term.
d) What term has a value of 61 ?

Solutions

a) Since the sequence is going up in threes,
the next three numbers are 13,16,19

b)

The rule for the nth term is 3n-2

c) The value of the 20th term is 58

d) The 21st term has a value of 61/

## Picture sequences

Example

Find the rule that connects the number of bars (b) to the number of dots (d) for the following sequence :

Notice that each time a dot is added, 2 bars are also added.

Instead of writing out the sequence with n underneath the term,
write it on top.

Then proceed as above.

So,

How many bars are needed if 52 dots are used ?

103 bars are required.

How many dots are needed if 51 bars are used ?

26 dots are required.

## Non constant difference

If the difference between the values is not a constant,
look for a constant (k) between the differences.
The expression now contains    ax2 ,
where a is the value of the new constant divided by 2.
Square each term, then multiple by ½k

Example

Find the rule for  the sequence    6,  18,   38,   66

Check: for n=2,
4n2+ 2 = 4x2x2 + 2 = 16 +2 = 18
Which is the value in the sequence