Question

How do you write a rule for the nth term 7,5,3,1,-1?

Answer 1

each term is going down by 2

`=> -2n`

this gives -2, -4, -6, -8

to get the sequence we need to find the adjuster

opposite of -2 is +2, 7+2=9

`-2n+9` or `9-2n`

Answer 2

`a_n=7-2n`

Explanation:

First of all, notice that this is an arithmetic sequence, i.e. two consecutive terms differ by a common difference. In this case, two consecutive terms always differ by `2`, which means that if you know the `n^"th"` terms, you will get the `n+1^"th"` by subtracting two.

We start from `a_0=7`, which is the starting point of the sequence. The next term, `a_1`, will be `a_0-2=7-2=5`, and so on.

The general rule is what we just described with words: start from the initial value `7`, and subtract `2` with each iteration. This means that, after `k` iterations, we will have subtracted two `k` times, i.e. we will have subtracted a total of `2k`.

So, the rule is

`a_n=7-2n`

You can confirm this by building some terms using the definition: given the starting value `a_0`, we have

`a_\color(red)(1) = a_0-\color(red)(1)*2`
`a_\color(red)(2) = a_1-2 = (a_0-2)-2 = a_0-\color(red)(2)*2`
`a_\color(red)(3) = a_2-2 = (a_0-2*2)-2 = a_0-\color(red)(3)*2`
`a_\color(red)(4) = a_3-2 = (a_0-3*2)-2 = a_0-\color(red)(4)*2`

as you can see, the index of the term is equal to the times we have to subtract `2`.