Help with sequences

Sequences

A sequence is list of numbers which can be worked out using a rule.

Each number in the sequence is called a term.

And each term has a position in the sequence.

The first term has position 1. The hundredth term has position 100.

The numbers in the sequence can either increase, go up, or decrease, go down.

Sequences can include negative numbers as well as positive numbers.

Each term can be generated from the previous term -- that is, term-to-term.

The term-to-term rule may be fixed, but the sequence will change if the first term changes.

Or, the term can be generated using a formula based on the position of the term.

Add on the same number

For example...

7, 11, 15, 19, 23... +4 on to each subsequent term

Subtract the same number

For example...

27, 22, 17, 12, 7, 2, -3, -8... -5 from each subsequent term

Multiply by the same number

For example...

2, 4, 8, 16, 32, 64, 128... multiply each subsequent term by 2

Divide by the same number

For example...

36, 12, 4... divide each subsequent term by 3

Add the two previous numbers

For example...

The rules for the terms of a sequence can also be written in terms of the position of the terms.

To do this you find the first difference between the terms, and in some cases the second difference.

For example... take the sequence 4, 7, 10, 13, 16, 19, 21...

The first difference is 7 - 4 = 3, or, 10 - 7 = 3...

Each number in the sequence has a position... the number 4 is the first in the sequence, 7 is second...

The difference between each term is +3.

In the first position, the term is 4. The position n = 1

The link between the position n = 1 and term is... 3n... 3 x 1... in the case of position 1, (3 x 1)... and then add 1 to get the term 4.

U_n = 3n + 1

So... the first term

U₁ = (3 x 1) + 1 = 4

To find,say, the 99^th term, n = 99

U₉₉ = (3 x 99) + 1 = 297 + 1 = 298