Question

Here are some values of sequence Q. Write a recursive definition for the sequence.

Answer 1

Answer:     $a_{1}$ = 3 ;  $a_{k}$ = $a_{k - 1}$ + 2.5, where k ≥ 2

Step-by-step explanation:

To define a sequence recursively, we must state the first term and then state a rule for how each successive term can be described from the one before it. For instance, suppose I had the sequence 5, 8, 11, 14, ...

The first term would be $a_{1}$ = 5

Each successive term where subscript k = 2 for the second term, 3 for the third, and so on, would be  $a_{k}$ = $a_{k - 1}$ + 3

So, for my example, the recursive definition for that sequence would be

       $a_{1}$ = 5 ;  $a_{k}$ = $a_{k - 1}$ + 3, where k ≥ 2

In the exercise you have, we go up 2 terms from the first to the third and the value goes up 5 units. We go up 4 terms from the third term to the 7th, we go up 10 units. Apparently, we go up 2.5 units in value as we go from one term to the very next one.

So ...

        $a_{1}$ = 3 ;  $a_{k}$ = $a_{k - 1}$ + 2.5, where k ≥ 2

This happens to be an arithmetic sequence, because we go up the same amount from one term to the next, but please do not assume that is  required for creating a recursive definition. It is not.

I hope this helps.