Arithmetic Sequences
Recursive and Explicit Formulas
Let’s look at the arithmetic sequence
20, 24, 28, 32, 36, . . .
This arithmetic sequence has a common difference of 4, meaning that we add 4 to a term in order to get the next term in the sequence.
The recursive formula for an arithmetic sequence is written in the form

For our particular sequence, since the common difference (d) is 4, we would write

So once you know the common difference in an arithmetic sequence you can write the recursive form for that sequence. ______
The explicit formula is also sometimes called the closed form. To write the explicit or closed form of an arithmetic sequence, we use

an is the nth term of the sequence. When writing the general expression for an arithmetic sequence, you will not actually find a value for this. It will be part of your formula much in the same way x’s and y’s are part of algebraic equations.
a1 is the first term in the sequence. To find the explicit formula, you will need to be given (or use computations to find out) the first term and use that value in the formula.
n is treated like the variable in a sequence. For example, when writing the general explicit formula, n is the variable and does not take on a value. But if you want to find the 12th term, then n does take on a value and it would be 12.
d is the common difference for the arithmetic sequence. You will either be given this value or be given enough information to compute it. You must substitute a value for d into the formula. You must also simplify you
formula as much as possible.
Examples:
  1. Write the explicit formula for the sequence:
20, 24, 28, 32, 36, . . .
The first term in the sequence is 20 and the common difference is 4. This is enough information to write the explicit formula.

Now we have to simplify this expression to obtain our final answer.

So the explicit (or closed) formula for the arithmetic sequence is .
Notice that an the and n terms did not take on numeric values. They are a part of the formula, again like x’s and y’s in algebraic expressions.
If we wanted to find the 50th term of the sequence, we would use n = 50. Look at the example below to see what happens.
  1. Given the sequence 20, 24, 28, 32, 36, . . . find the 50th term.
To find the 50th term of any sequence, we would need to have an explicit formula for the sequence. Since we already found that in Example #1, we can use it here. If we do not already have an explicit form, we must find it first before finding any term in a sequence.
Use the explicit formula and let n = 50. This will give us