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Question

Enter a recursive rule and an explicit rule for the arithmetic sequence. Then, find the 20th term of the sequence.

2, 0, −2, −4, ...

The recursive rule is f(1) = _______ f(n) = f (n+1 or n-1 or n) -______

The explicit rule is f(n) =_____ - ______ (n+1 or n-1 or n)

f(20) = ______

Answer 1

The above sequence is an arithmetic sequence where the next term is the previous term minus 2.

The first term also known as f(1) = 2

The recursive rule states that the next term is the previous term minus 2. This can be written as...

f(n) = f(n-1) -2

f(n-1) is the term just before f(n)

The explicit rule is in the form.

f(n) = f(1) + (n-1)*difference.

The difference is -2 and the starting term is 2 so it can be written as....

f(n) = 2 + (n-1)*(-2)

To find a specific term (in this case the 20th term plug in 20 for n.

f(20) = 2 + (20-1)*(-2)

f(20) = -36

Answer 2

as the first term in the sequence is 2, f(1) = 2. You can also see that each term decreases by two (2 to 0, 0 to -2, and so on...).

So the recursive rule is

f(n) = f(n-1) - 2,

as you are subtracting 2 from the previous term each time.

then the explicit rule is then...

f(n) = 2 - 2(n-1)

Using this rule, f(20) is equal to 2 - 2*19 = -36.