GCSE(F), GCSE(H),

The value of a given term of a linear sequence can be worked out using the **general term** of the sequence. The general term is also known as the **nth term**, because, for any term *n* in the sequence, the value of the term can be worked out.

To work out what the *n*th term of a sequence is, work out the common difference. Given the sequence 4, 7, 10, 13, ..., work out the difference between each value:

Term | 1 | 2 | 3 | 4 | ... | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Value | 4 | 7 | 10 | 13 | ... | ||||||

Difference | 3 | 3 | 3 | ... |

Next, find out the value of the zeroth term, by subtracting the difference from the value of the first term:

Term | 0 | 1 | 2 | 3 | 4 | ... | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Value | 1 | 4 | 7 | 10 | 13 | ... | |||||

Difference | 3 | 3 | 3 | 3 | ... |

Any term can now be worked out:

*difference* x *term* + *zeroth term value*.

The nth term is 3*n* + 1 (3 is the difference, and 1 is the value of the zeroth term).

For any term, its value can be worked out by substituting the *n* with the term number.

For the value of the 30th term in the sequence 3*n* + 1: replace *n* with 30 and calculate 3 x 30 + 1 = 91.

1. What is the general term (nth term) for the sequence 2, 0, -2, -4?

Answer: -2n + 4

Set out a table:

Term | 0 | 1 | 2 | 3 | 4 | ... | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Value | 4 | 2 | 0 | -2 | -4 | ... | |||||

Difference | -2 | -2 | -2 | -2 | ... |

The difference is -2, and the value of the *n*term is 4. The nth term is -2n + 4

2. Is 32 a number in the sequence given by 3*n* - 5?

Answer: No

Set the question as an equation: 32 = 3n - 5.

37 = 3n (added 5 to both sides)

12.33 = n (divided both sides by 3)

Because 12.33 is not a whole number it cannot be a term; therefore 32 is not in the sequence.

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