Maths

Patterns and Sequences

L . O to find the n^{th} term of a shape and arithmetic sequence.

A sequence is a patterns that involves shapes or numbers that follow a rule.

These online methods will help you tackle questions with ease and get the

best grades!

Differences of consecutive terms increase by 1 each time so in the next

term you would +5, +6, +7 and so on. The 5th term would be 10+5 =15.

The 6th term would be 15 + 6 = 21.

Also notice these are all triangular numbers.

Differences of consecutive terms increase by 1 each time however in the

next term you would +4, +5, +6 and so on. The 5th term would be 6+4 =10.

The 6th term would be 10 +5 = 15.

Differences of consecutive terms do not increase in a linear fashion,

however notice that the total number are all square numbers.

The 5th term would be 52 = 25

The 6th term would be 62 = 36

In general the nth term of the sequence is denoted by n2.

10th term = 102 = 100

100th term = 1002 = 10000

nth term of an arithmetic series

An arithmetic sequence is where you apply the same rule to each term, which

maybe the difference, or a 2 step rule which involves multiplying by a constant

and then adding another constant.

Example 1

Here is an arithmetic sequence 5, 8, 11, 14, 17. Find:

- The difference between consecutive terms
- The zero term
- An expression to find the nth term

1.

Term Number (n) | Output |

1 | 5 |

2 | 8 |

3 | 11 |

4 | 14 |

5 | 17 |

The difference is 3

Step 1:

Find the difference between term 2 and term 1.

8 – 5 = 3

Step 2:

Find the difference between term 3 and term 2.

11 – 8 = 3

This step is just to confirm the difference is always the same throughout.

2. The zero term can be found by working backwards. In the sequence if you

add 3 to the output you gain the next term. Here instead of adding 3,

subtract 3 from the first term.

5– 3 = 2 the 0th term

3. Before tackling this part, there is one simple rule you need to know. To

find the nth term of a sequence:

nth term = difference x n + zero term

nth term = 3x n + 2

Check: Term 5 = 17

(3 X 5 ) + 2 = 17!!

The nth term = 3n + 2

Step 1:

Note down the difference and the 0th term (if not done so already and apply

it to the rule )

Step 2:

Double check the nth term you’ve calculated works using an existing term.

Step 3:

Remember to remove any multiplication signs because this is an algebraic

expression.

Related Topics

simultaneous-equations changing-the-formulae

Adding-subtracting-formulas Algebraic-proof