Question

How do you write the rule for the nth term given `1/4,2/5,3/6,4/7,5/8,...`?

Answer 1

`a_n = n/(n+3)`

Explanation:

The way the sequence has been written seems to indicate an intention that the rule for the general term is:

`a_n = n/(n+3)`

Both the numerators (`1, 2, 3, 4, 5`) and denominators (`4,5,6,7,8`) are arithmetic sequences with common difference `1`.

Note that the same sequence of values could have been written:

`1/4, 2/5, 1/2, 4/7, 5/8,...`

The same formula would be correct, but slightly harder to spot.

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Footnote

Note that unless you are told what kind of sequence you are dealing with then any finite initial sequence of values does not determine a unique formula for an infinite sequence.

For example, consider the sequences:

`1/3, 2/9, 3/27,...`

`3/9, 2/9, 1/9,...`

These are both the same sequence, but the way they are expressed would lead you to different conclusions about the following term and general formula.

In the first case, you would probably deduce that the next term is `4/81` and general formula: `a_n = n/3^n`.

In the second case, you would probably deduce that the next term is `0/9` and general formula: `a_n = (4-n)/9`