Find the recursive rule, explicit rule, and f(20)

Recursive:

Explicit:

And the 20th term f(20) is -72.

Step-by-step explanation:

We have the sequence:

4, 0, -4, -8...

Let’s find the recursive and explicit rule for the sequence.

Recursive Rule:

Let’s determine the factor by which the sequence is decreasing by. We see that each subsequent term is 4 less than the previous term. In other words, our common difference is -4.

So, this is a arithmetic sequence.

The standard format for an recursive rule for an arithmetic sequence is:

Where a is the initial term and d is our common difference.

From our sequence, we know that our initial term a is 4.

We also determined that our common difference is -4.

Substitute. Hence, our recursive rule is:

Explicit Rule:

The standard format for an explicit rule for an arithmetic sequence is:

Where a is the initial term and d is the common difference. So, again, let’s substitute 4 for a and -4 for d. Hence, our explicit formula is:

To find f(20), we can use the explicit formula. It is possible to use the recursive formula, but it gets tedious. Therefore, we will substitute 20 for n for our explicit formula:

Evaluate:

Hence, our 20th term is -72.

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for

28

step-by-step explanation:

because that's the answer