Sequences

We can express this as a recursive formula by writing:

This says to get any term in the sequence (a_n), add one (+1) to the previous term (a_n-1).

This is a recursive formula because each successive term relies on the term before it. In other words, the 5th term relies on knowing the 4th term, and the 4th term relies on knowing the 3rd term, and so on. This is how we naturally find sequences in our heads, but is not very helpful when we need to find something like the 50th term!

There is an easier way to find larger number terms, such as the 50th term, and that is the explicit formula. With the explicit formula, we simply plug in the term number we want to find, and the output is the term in the sequence. Lets look at the explicit formula for our simple example above:

a_n 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.

Lets do an example.

Write the recursive and explicit formula for the sequence:

20, 24, 28, 32, 36, . . .

A geometric sequence is like an arithmetic sequence, except that instead of a common difference between the terms, it has a common ratio. Lets look at an example:

This geometric sequence has a common ratio of 3, meaning that we multiply each term by 3 in order to get the next term in the sequence.

The recursive formula for a geometric sequence is written in the form

For our particular sequence, since the common ratio (r) is 3, we would write

Once you know the common ratio in a geometric sequence you can write the recursive form for that sequence.

What about and explicit formula?

a_nis the nth term of the sequence. When writing the general expression for a geometric 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.

r is the common ratio for the geometric sequence. You will either be given this value or be given enough information to compute it. You must substitute a value for r into 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.

Write the recursive and explicit formula for the sequence:

2, 6, 18, 54, 162, . . .

What if we are not given the sequence, but rather the following:

Find the explicit formula for a sequence where r = 2 and .

We can use a little algebra to figure this out!