Write the recursive formula for this geometric sequence. 8,-8,8,-8...

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lemjay's profile pic

lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

To determine the recursive formula of the geometric sequence 8, -8, 8, -8, we have to know the number that should be multiplied to the previous term to get the next term.

Notice that in the given sequence, only the sign of 8 changes. So,  let's check if multiplying the previous term by -1 is the rule.

`a_1=8`

`a_2=8*(-1)=-8`

`a_3=-8*(-1)=8`

`a_4=8*(-1)=-8`

This verifies that multiplying the previous term (a_(n_1)) by -1 is the rule to get the next term (a_n).

So the pattern is:

`a_1=8`

`a_2=a_1*(-1)=-a_1`

`a_3=a_2*(-1)=-a_2`

`a_4=a_3*(-1)=-a_3`

Therefore, the recursive formula is `a_n= -a_(n-1)` .

chrisyhsun's profile pic

chrisyhsun | Student, College Freshman | (Level 1) Salutatorian

Posted on

Before writing a formula for any type of sequence, it is important that you can in some way articulate what the pattern is, even if it’s not in technical language yet. For example, looking at the sequence in the problem, you must first see that it flip-flops between 8 and -8.

After identifying what the pattern is, the next step is thinking of why this might be the case. This is a somewhat trial-and-error process. For example, you might think – oh, all I have to do is subtract 16 to go from the first to second number, then add 16 from the second to the third, then repeat this pattern. While this is a valid strategy, you can’t represent this in a geometric sequence, so you need to start thinking of other approaches, such as multiplying by -1 each time. 

Once you have the methodology down, all you have to do is write it down. Geometric sequences are written in the form: A(n) = a*b^n. You can define n as starting at either 0 or 1. b is the value that you are multiplying by each time, or -1 for this problem. a is used as a “correction factor” of sorts, making the flip-flopping pattern actually fit the specific sequence, and a will be 8 for our problem. Let’s start n at 0, in which case the geometric sequence is represented as: A(n) = 8 * (-1)^n.

•Final point to keep in mind: in my response, I tried to walk through the steps on solving sequences in general. It is useful to remember, however, that any sequence that flip flops between negatives and positives will mostly use a negative as the b, so that will probably save you some thinking in the future!

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