Question

3) Here is a recursive function for sequence f: f(1)=23,f(n)=f(n−1)−7. The explicit formula for the term is f(n)=23−7(n−1). a. Explain how you know that these definitions represent the same sequence. b. Select a definition to calculate f(20), and explain why you chose it.

Answer 1

Using the recursive and explicit formula

first three terms are 23,16, 9 and it is same for both . So they have same sequences .

Recursive function can be applied using previous term . So we use Explicit formula to find f(20)

f(20)=-110

Given :

Recursive function for sequence f: f(1)=23,f(n)=f(n−1)−7

explicit formula for the term is f(n)=23−7(n−1).

Lets find out first three terms using both the formulas

Recursive function , Replace n=2,3 to find out second and third terms

$f(1)=23,f(n)=f(n-1)-7\\n=2\\ f(2)=f(2-1)-7\\f(2)=f(1)-7=23-7=16\\n=3\\f(3)=f(2)-7=16-7=9$

So first three terms are 23,16, 9

Lets do the same using explicit formula but we take n=1,2,3

$f(n)=23-7(n-1)\\n=1\\f(1)=23-7(1-1)=23\\f(2)=23-7(2-1)=16\\f(3)=23-7(3-1)=9$

First three terms using explicit formula is 23,16,9

From these we can see that  these definitions have same sequence.

Now we find out f(20). To find f(20) we use explicit formula because to use recursive function we need f(19) to find f(20)

Recursive function can be applied using previous term

$f(n)=23-7(n-1)\\n=20\\f(20)=23-7(20-1)\\f(20)=-110$

The value of f(20)=-110

Learn more : brainly.com/question/17139881