List the first five terms of the arithmetic sequence whose recursive rule is {a_n} = (a_{n – 1}) – 9 and whose first term is 72.

First off, 72 is the first term as defined by the problem.

a₁ = 72

Subtract 9 from 72 to get the second term.

a₂ = 72 – 9 = 63

Continue this process to find the third, fourth, and fifth terms.

a₃ = 63 – 9 = 54
a₄ = 54 – 9 = 45
a₅ = 45 – 9 = 36

Sweet. Our first five terms are:

72, 63, 54, 45, 36

Find the recursive rule for the sequence:

Use simple addition or subtraction to find the common difference, d.

Looks like we're adding one-third over and over to get each new term. Now all we've gotta do is plug that into our general form for the recursive rule of an arithmetic sequence.

{a_n} = (a_{n – 1}) + d

Find the explicit rule for the arithmetic sequence:

-17, -6, 5, …

We want an explicit rule this time, so we'll need the first term in the sequence.

a₁ = -17

What's the common difference?

-17 + 11 = -6
-6 + 11 = 5

Yup, we're adding 11 each time, so d = 11.

Now we can plug the values for a₁ and d into our general formula.

{a_n} = a₁ + d(n – 1)
{a_n} = -17 + 11(n – 1)
{a_n} = -17 + 11n – 11
{a_n} = 11n – 28

Given the arithmetic sequence  with a₁ = 4, what's the 100th term?

We want the 100th term, and we really don't wanna subtract that fraction 100 times in a row if we can help it.

We need to get to an explicit rule. That means we need the first term and the common difference. We know a₁ = 4, but what about d?

The problem started us out with a recursive rule. These guys always follow the form {a_n} = (a_{n – 1}) + d. So d is just the fraction tacked onto the end.

Use this info to put together an explicit rule using the general form.

{a_n} = a₁ + d(n – 1)

Plug in the term number needed, in this case 100, for n in your formula.

How many total terms are there in the sequence 1, 6, 11, … 91, 96?

Locate the common difference, first term, and last term of the sequence.

d = 5
a₁ = 1
a_n = 96

Use the general rule of an explicit arithmetic sequence, and pop in all the values we know. That gives us an equation with one variable: the number of terms, n.

a_n = a₁ + d(n – 1)
96 = 1 + 5(n – 1)

Finish up by solving for n, which tells us the term number.

96 = 1 + 5(n – 1)
96 = 1 + 5n – 5
96 = 5n – 4
100 = 5n
20 = n