Geometric Sequence Defined
Geometric sequence: a set of numbers in numerical order, with a non-zero common ratio between each term.
A sequence is a set of numbers in numerical order. You may be familiar with arithmetic sequences, which have a common difference between each term, a constant number that is either added or subtracted as we go from one term to the next. With geometric sequences, there is a common ratio between each term.
Common ratio: the ratio between any two consecutive terms in a geometric sequence, a constant value.
Similar to arithmetic sequences, the common ratio is a constant value; however it is not added or subtracted from one term to the next, it is multiplied by the preceding term to continue the sequence.
Consider the following sequence:
This is an example of a geometric sequence with a common ratio of 2. We can multiply a term by 2 to get the value of the next term. The next term in the sequence would be 96, and 192 would follow that.
Divergent and Convergent Sequences
The previous sequence was also an example of a divergent sequence.
Divergent sequence: a sequence whose terms do not have a finite limit; they tend toward positive or negative infinity.
Since we are continuously multiplying each term in the sequence by 2 in the above example, we know that the terms will continue to increase in value, making the limit of values positive infinity. Even if the common ratio were –2, the terms would alternate between positive and negative, but would still grow in absolute magnitude. Divergent sequences are characterized by a common ratio greater than 1 or less than –1.
In contrast, some geometric sequences are convergent. The terms in convergent sequences do have a limit.
Convergent sequence: a sequences whose terms have a finite limit; they tend toward a specific value.
Consider if we have geometric sequence with a common ratio of 0.5, as in the example below:
In this sequence, each term is being cut in half as the sequence continues. Eventually, the term would be virtually zero, thus the sequence is convergent. Even if the common ratio were –0.5, the terms would still converge to zero, even though they would alternate between positive and negative. Convergent sequences are characterized by a common ratio in between –1 and 1.
Divergent and convergent sequences are characterized by their common ratio, r:
The Formula for Geometric Sequences
Let's return to the geometric sequence:
It was easy enough to find the value of the next two terms, by multiplying 48 by 2 to get 96, and then by 2 again to get 192. What if we wanted to find the value of the 78th term? Certainly finding the value of that term by continuously multiplying by 2 is inefficient. We can use the following formula:
where:
Writing a Formula for a Geometric Sequence
Consider the following geometric sequence:
How can we develop a formula to represent this sequence, so that we may use it to find the value of any term?
The value of the first term is 8748, so this will be a_{1} in our formula. We need to calculate the common ratio. To do so, we will take any two consecutive terms, and divide the second by the first: 324 ÷ 972 = 1/3. This tell us that each term is multiplied by 1/3 to get the value of the next term.
The formula for this sequence is:
Using the Formula to find a Term
Let's use the above formula to find the value of the 7th term in the sequence. Since we want to find the 7th term, we use n = 7 in our formula:
The ratio between any two consecutive terms in a geometric sequence; a constant value
A sequence whose terms have a finite limit; they tend toward a specific value
A sequence whose terms do not have a finite limit; they tend toward ± infinity
A set of numbers in numerical order, with a non-zero common ratio between each term