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

2 Answers
Jun 27, 2018

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

Jun 27, 2018

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.