Sequences

The Basics

Sequences are sets of numbers that follow a rule or a pattern.

If this rule is to add or subtract a number, it is an arithmetic sequence, while if it is to multiply or divide by a number, it is a geometric sequence.

Term-to-term Rules

Term-to-term rules define how to get from one term to another. For example, with a sequence where the next number is 5 greater than the last, the rule is ‘add 5’.

nth Term Rules

$n$^th term rules define a formula to take a value ($n$) which represents the ‘position’ in the sequence and find which number falls in that position in the sequence.

Arithmetic nth Terms

To find the $n$^th term of an arithmetic sequence, simply find the difference between two terms ($a$) and then take this value away from the first term ($b$). Then, use this format:

$$an+b$$

Geometric nth Terms

To find the $n$^th term of a geometric sequence, first, work out the difference between each value. This difference will not be the same. Next, you must find the difference between these differences ($a$). Then, find the value in the sequence below the first by taking a from the difference between the first and second term, then take that from the first term. This is $b$. Apply this format:

$$\frac{a}{2}{n}^2+b$$