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Linear Number Sequences

Document Reference: TN201402003 - Rev: 4.3 - Last Update: 19-03-2014 13:54 GMT - Downloaded: 22-Jun-2024 14:57 GMT

A linear number sequence (aka arithmetic sequence) is an ordered set of numbers which has a common (constant) difference between consecutive terms.

Consider this linear number sequence: 2, 9, 16, 23, 30, 37, 44, 51, ....

Terms in a linear number sequence.


Each number in a sequence, separated by commas, is called a term of this sequence. The first term is called T1, followed by T2, T3, and so on.

Term Value

In a linear number sequence, the term value is the number value of one term. In above example the term value of T6 = 37.

Common Difference

The value added between each consecutive term is called the common difference and is always constant in any linear number sequence. We use the letter d to represent the common difference. The common difference can be a positive or negative value.

In above example, the common difference equals 7.

Finding the Next Term

The rule for finding the next term is to add d (common difference) to the previous term:

Tn+1 = Tn + d.

In above example, the rule for finding the next term is to add 7 (common difference) to the previous term, e.g.

T9 = T8 + 7         T9 = 51 + 7         T9 = 58.

Finding the nth Term

The nth term (Tn) of a linear number sequence allows to calculate any term of the sequence:

Tn = d × n + T1 - d.

We can now work out any term for above example. E.g. the 40th therm:

T40 = 7 × 40 + 2 - 7         T40 = 280 + 2 - 7         T40 = 275.

Graphing Linear Number Sequences

We can graph the term on the x axis and the term value on the y axis. Joining all the points will result a linear (straight) line.

Graphing above example:

Graphing terms in a linear number sequence.

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