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

Answer:

#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#.