Sequences, Series and Binomial Theorem
Geometric Sequences and Series
Learning Objectives
By the end of this section, you will be able to:
- Determine if a sequence is geometric
- Find the general term (nth term) of a geometric sequence
- Find the sum of the first terms of a geometric sequence
- Find the sum of an infinite geometric series
- Apply geometric sequences and series in the real world
Before you get started, take this readiness quiz.
Determine if a Sequence is Geometric
We are now ready to look at the second special type of sequence, the geometric sequence.
A sequence is called a geometric sequence if the ratio between consecutive terms is always the same. The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.
A geometric sequence is a sequence where the ratio between consecutive terms is always the same.
The ratio between consecutive terms, is r, the common ratio. n is greater than or equal to two.
Consider these sequences.
Determine if each sequence is geometric. If so, indicate the common ratio.
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ⓑ
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To determine if the sequence is geometric, we find the ratio of the consecutive terms shown.
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ⓑ
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Determine if each sequence is geometric. If so indicate the common ratio.
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ⓑ
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ⓐ The sequence is geometric with common ratio ⓑ The sequence is geometric with common ratio ⓒ The sequence is not geometric. There is no common ratio.
Determine if each sequence is geometric. If so indicate the common ratio.
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ⓑ
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ⓐ The sequence is not geometric. There is no common ratio. ⓑ The sequence is geometric with common ratio ⓒ The sequence is geometric with common ratio
If we know the first term, and the common ratio, r, we can list a finite number of terms of the sequence.
Write the first five terms of the sequence where the first term is 3 and the common ratio is
We start with the first term and multiply it by the common ratio. Then we multiply that result by the common ratio to get the next term, and so on.
The sequence is
Write the first five terms of the sequence where the first term is 7 and the common ratio is
Write the first five terms of the sequence where the first term is 6 and the common ratio is
Find the General Term (nth Term) of a Geometric Sequence
Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence.
Let’s write the first few terms of the sequence where the first term is and the common ratio is r. We will then look for a pattern.
As we look for a pattern in the five terms above, we see that each of the terms starts with
The first term, is not multiplied by any r. In the second term, the is multiplied by r. In the third term, the is multiplied by r two times ( or ). In the fourth term, the is multiplied by r three times ( or ) and in the fifth term, the is multiplied by r four times. In each term, the number of times is multiplied by r is one less than the num