This lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence. In this lesson, it is assumed that you know what an arithmetic sequence is and can find a common difference. If you need to review these topics, click here.
2, 6, 18, 54, 162, . . .
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
So once you know the common ratio
in a geometric sequence
you can write the recursive form for that sequence.
However, the recursive formula can become difficult to work with if we want to find the 50th term. Using the recursive formula, we would have to know the first 49 terms in order to find the 50th. This sounds like a lot of work. There must be an easier way. And there is!
Rather than write a recursive formula, we can write an explicit formula. The explicit formula is also sometimes called the closed form. To write the explicit or closed form of a geometric sequence, we use
is 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.
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.
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.
Your formulas should be simplified if possible, but be very careful when working with exponential expressions. We’ll look at this more closely in the examples.
- Let’s go back and look at the sequence we were working with earlier and write the explicit formula for the sequence.
2, 6, 18, 54, 162, . . .
This is enough information to write the explicit formula.
Be careful here! DO NOT multiply the 2 and the 3 together. Order of operations
tells us that exponents are done before multiplication. So 3 must be raised to the power as a separate operation
from the multiplication.
So the explicit (or closed) formula for the geometric sequence
Notice that the an 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 10th term of the sequence, we would use n = 10. Look at the example below to see what happens.
- Given the sequence 2, 6, 18, 54, 162, . . . find the 10th term.
To find the 10th term of any sequence, we would need to have an explicit formula for the sequence. Since we already found that in our first example, 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 = 10. This will give us
Notice how much easier it is to work with the explicit formula than with the recursive formula to find a particular term in a sequence.
What happens if we know a particular term and the common ratio, but not the entire sequence? Let’s see in the next example.
- Find the explicit formula for a sequence where r = 2 and .
The formula says that we need to know the first term and the common ratio. We have r, but do not know a1
. However, we have enough information to find it. We know that when n = 12, the 12th
term in the sequence
If we simplify that equation, we can find a1.
Now that we know the first term along with the r value given in the problem, we can find the explicit formula.
Notice this example required making use of the general formula twice to get what we need. The first time we used the formula, we were working backwards from an answer and the second time we were working forward to come up with the explicit formula.
- Find the explicit formula for a geometric sequence where and .
In this situation, we have the first term, but do not know the common ratio. However, we do know two consecutive terms which means we can find the common ratio
Now we use the formula to get
Notice that writing an explicit formula always requires knowing the first term and the common ratio. If neither of those are given in the problem, you must take the given information and find them.