Geometric Sequences


A Geometric Sequence is a sequence of numbers that has the property that the ratio between two consecutive elements is constant, equal to a certain value \(r\). This value is also known as the common ratio.

Assume that the first term is \(a\). Then, the next term is \(a r\), and the next is \(ar^2\). And so on.

So, in other words, we start with the first term \(a\), and the next term is always found by multiplying the previous term by \(r\).

So the first term is \(a_1 = a\).

The second term is \(a_2 = a r\).

The third term is \(a_3 = a r^2\).

Therefore, the general nth term is

\[\large a_n = a r^{n-1}\]

Summarizing, you need two pieces of information to define a geometric sequence: You need the initial term \(a\), and the constant ratio \(r\). Consecutive terms of the sequence are obtained by multiplying the previous term by \(r\).

EXAMPLE 1: Example of a geometric sequence

Find the 6th term of a geometric sequence with initial term \(10\), and \(r = 1/2\).

ANSWER:

Based on the information provided, we have enough information to define the geometric sequence. Indeed, we have the first term \(a = 10\), and we have the constant ratio \(r = 1/2\).

The general nth term is

\[\large a_n = a r^{n-1}\]

so then the 6th term is

\[\large \displaystyle a_{10} = a r^{6-1} = 10 \left(\frac{1}{2}\right)^5 \] \[\large = \frac{10}{32} = \frac{5}{16} \]

Can the Common Ratio be Negative?

Yes, absolutely. The constant ratio \(r\) can be negative. For example, we can have a geometric sequence with initial term \(a_1 = 1\) and constant ratio \(r = -2\). So then, the second term is \(a_2 = 1 \cdot (-2) = -2\), \(a_3 = (-2) \cdot (-2) = 4\), and so on.

So, it is the exactly the same rule: in order to the following term, we multiply the previous term by the constant ratio \(r\), even if the constant ratio is negative.

EXAMPLE 2

Find the 5th term of a geometric sequence with initial term \(3\), and \(r = -2\).

ANSWER:

We have enough information to define the geometric sequence, because we have the first term \(a_1 = 3\), and we have the constant ratio \(r = -2\).

The general nth term (with negative constant ratio) is

\[\large a_n = a r^{n-1} = 3 \cdot (-2)^{n-1}\]

so then the 5th term is

\[\large \displaystyle a_{5} = a r^{5-1} = 3 \cdot (-2)^{5-1} = 3 \cdot (-2)^4 = 3 \cdot 16 = 48\]

EXAMPLE 3

Consider the sequence 1, 1/2, 1/4, 1/16, ... Is this sequence geometric?

ANSWER:

In order for a given sequence to be geometric, the terms need to have a common ratio. In this case, dividing the second term by the first term we get \((1/2)/1 = 1/2\).

Then, if we divide the third by the second term: \((1/4)/(1/2) = 1/2\). So far so good.

Now, if we divide the fourth by the third term: \((1/16)/(1/4) = 1/4\). It fails. It is not a geometric series, because it does not have a common ratio (the ratio is 1/2 for the first two terms, but then it is 1/4, so it is not constant).

Hence, the sequence is NOT a geometric sequence.


More About the Geometric Sequences

The punchline you need to keep in mind. What is the formula of geometric sequence? Simple

\[\large a_n = a r^{n-1}\]

where \(a\) is the initial term and \(r\) is the constant ratio (or common ratio, as it is also called).

There are a couple of calculators that you may want to use that are related to the concept of geometric sequence, or geometric progression, as it is also called..

• First you can check our infinite geometric series sum calculator, which sums infinite terms of a geometric sequence. This sum will be well defined (converge) if the constant ratio is such that \(|r| < 1\).

• Also, you will want to use our geometric sequence sum calculator, which computes the sum of terms in a geometric sequence, UP to a certain finite value. This sum is well defined without conditions on the constant ratio \(r\), provided that we add up to a finite term of the sequence.

Can a geometric sequence have a common ratio of 1?

Absolutely. The general term for a geometric sequence with a common ratio of 1 is

\[\large a_n = a r^{n-1}= a \cdot 1^{n-1} = a\]

So, a sequence with common ratio of 1 is a rather boring geometric sequence, with all the terms equal to the first term.




In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us.

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