Geometric sequences are used in mathematics whenever we have a sequence of numbers that grows or shrinks by a fixed percentage at each step. However, there are a few things you should know about these sequences.

So, what is a geometric sequence? **A geometric sequence is a sequence of numbers that increases or decreases by the same percentage at each step. The ratio between consecutive terms in a geometric sequence is always the same. This ratio r is called the common ratio, and the nth term of a geometric sequence is given by a _{n} = ar^{n}.**

Of course, a geometric sequence can have positive or negative terms. The common ratio r can also be positive or negative.

In this article, we’ll talk about geometric sequences and answer some common questions about them. We’ll also look at some examples to make the concept clear.

Let’s get started.

## What Is A Geometric Sequence?

A geometric sequence (or geometric progression) is a sequence of numbers that increases or decreases by the same percentage at each step. That is, the ratio between two consecutive terms in a geometric sequence is always the same.

This ratio, r, is called the common ratio of the geometric sequence. If the nth term of a geometric sequence is a_{n}, then the common ratio r is:

**r = a**_{n+1}/ a_{n}

Note that this formula also tells us how to find the next term in the sequence from the previous term. If we solve for a_{n+1}, we get:

**a**_{n+1}= a_{n}r

We can use any nonnegative value of n to find the value of r. For example, if we choose n = 0, then we only need the first two terms of the geometric sequence to find r:

**r = a**_{1}/ a_{0}

**Note that according to this convention, the term with index n = 0 is the first term – we see this often in computer science as well.**

If we choose n = 10, we would need the 10^{th} and 11^{th} terms of the geometric sequence to find r:

**r = a**_{11}/ a_{10}

### How To Find The Common Ratio Of A Geometric Sequence

If we have two consecutive terms in a geometric sequence, we can simply take their ratio (quotient) to find the common ratio r (as we did in the examples above). Just remember to divide in the correct order, since division is not commutative.

For example, given a_{1} = 8 and a_{2} = 32, we would calculate:

**d = a**_{2}/ a_{1}**=32 / 8****=4**

The common ratio for this geometric sequence would be r = 4. It would be incorrect to calculate 8 / 32 = 1 / 4.

In general, the term with the larger index (the number in the subscript) comes first (it is the dividend or numerator), and the term with the smaller index is the divisor or denominator.

If we have two terms that are not consecutive, we need to divide them (again, in the proper order) and then take a root (the correct root is given by the difference between the larger index and the smaller index).

So, given two terms a_{m} and a_{n} in a geometric sequence (with m < n), we would find the common ratio r with the formula:

**r = (a**_{n}/ a_{m})^{1/(n-m)}

For example, given the two terms a_{3} = 8 and a_{12} = 4096 in a geometric sequence, we would use the formula above to calculate d with m = 3, n = 12, a_{m} = 8, and a_{n} = 4096:

**r = (a**_{n}– a_{m}) / (n – m)**r = (a**_{12}/ a_{3})^{1/(12-3)}**r = (4096 / 8)**^{1/9}**r = (512)**^{1/9}**r = 2**

So, the common ratio for this geometric sequence is r = 2. With this information, we can work backwards to find the first term a_{0} if we want to:

**a**_{3}= 8**a**[since a_{2}= 4_{3}= a_{2}r, with a_{3}= 8 and r = 2]**a**[since a_{1}= 2_{2}= a_{1}r, with a_{2}= 4 and r = 2]**a**since a_{0}= 1 [_{1}= a_{0}r, with a_{1}= 2 and r = 2]

We can find a general formula for the first term a_{0} by solving the equation from before with m = 0:

**r = (a**_{n}/ a_{m})^{1/(n-m)}**r = (a**[use a = 0]_{n}/ a_{0})^{1/(n-0)}**r = (a**_{n}/ a_{0})^{1/n}**r = (a**_{n}/ a_{0})^{1/n}**r**raise both sides to the power of n]^{n}= a_{n}/ a_{0}[**r**[divide both sides by a^{n}/ a_{n}= 1 / a_{0}_{n}]**a**[take the reciprocal of both sides]_{n}/ r^{n}= a_{0}

So, given the nth term a_{n} and the common ratio r of a geometric sequence, we can find the first term a_{0} by the equation **a _{0} = a_{n} / r^{n}.**

#### Does A Geometric Sequence Always Increase?

A geometric sequence does not always increase. For example, if the first term a_{0} is positive and the common ratio r is positive and less than 1, the geometric sequence will decrease.

Take a_{0} = 8 and r = 1/4. Then the first several terms of the sequence are 8, 2, 1/2, 1/8, 1/32, … which are decreasing terms.

Keep in mind that a geometric sequence may not always increase or decrease. A geometric sequence can also oscillate between positive and negative values.

If the common ratio r is negative, then the terms of the geometric sequence will switch between positive and negative.

Take a_{0} = 1 and r = -2. The first several terms of the sequence are 1, -2, 4, -8, 16, -32, 64, … which switch between positive and negative.

#### Does Every Geometric Sequence Converge?

A geometric sequence does not always converge. A geometric sequence converges if |r| < 1 (that is, -1 < r < 1).

The reason is that as the index n increases, the nth term a_{n} approaches zero (since r^{n} approaches zero when |r| < 1).

If r = 1, the geometric sequence converges to a_{0} (a ratio of r = 1 means all the terms have the same value of a_{0} and the sequence converges to a_{0}). The terms would be a_{0}, a_{0}, a_{0}, a_{0}, …

If r = -1, the geometric sequence oscillates between a_{0} and –a_{0}. The terms would be a_{0}, -a_{0, }a_{0}, -a_{0}, …

If |r| > 1, the geometric sequence does not converge, since the terms will get larger in absolute value (get further away from zero) as n increases.

### How To Find The Nth Term Of A Geometric Sequence

To find a general formula for the nth term of a geometric sequence, we need the first term a_{0} and the common ratio r.

We can use the equation for a_{0} we found above and solve for a_{n} instead of a_{0}:

**a**[equation from above]_{n}/ r^{n}= a_{0}**a**[multiply by r_{n}= a_{0}r^{n}_{n}on both sides]

So, given the 1st term a0 and the common ratio r of a geometric sequence, we can find the nth term a_{n }by the equation **a _{n} = a_{0}r^{n}_{.}**

#### Example: How To Find The Nth Term Of A Geometric Sequence

Let’s say we are given the terms a_{5} = 486 and a_{8} = 13,122 of a geometric sequence. First, we will find r by the equation from earlier:

**r = (a**_{n}/ a_{m})^{1/(n-m)}**r = (13,122 / 486)**^{1/(8-5)}**r = (27)**^{1/3}**r = 3**

Now we need to find a_{0}. We will use the formula for a_{0} that we found earlier, along with a_{5} = 486:

**a**_{0}= a_{n}/ r^{n}**a**[use n = 5]_{0}= a_{5}/ r^{5}_{ }**a**_{0}= 486 / 3^{5}_{ }[a_{5}= 486 and r = 3]**a**_{0}= 486 / 243**a**_{0}= 2_{ }

Now we can find the formula for the nth term a_{n} by the equation from earlier:

**a**_{n}= a_{0}r^{n}**a**[a_{n}= 2*3^{n}_{0}= 2, r = 3]

We can check this formula for the 3 terms we already have to confirm:

**a**_{0}= 2*3^{0}= 2*1 = 2 (correct!)**a**_{5}= 2*3^{5}= 2*243 = 486 (correct!)**a**_{8}= 2*3^{8}= 2*6,561 = 13,122 (correct!)

So, the formula for the nth term of this geometric sequence is **a _{n} = 2*3^{n}.**

The first several terms of the geometric sequence are 2, 6, 18, 54, 162, 486, 1,458, …

### What Is The Recursive Formula For A Geometric Sequence?

The recursive formula for a geometric sequence is the one involving the next term, the previous term, and the common ratio r (the same one we saw earlier):

**a**_{n+1}= a_{n}r

It is called recursive because it refers to a previous term to find the next term.

### Can Geometric Sequences Be Negative?

Geometric sequences can have negative terms. In fact, every term of a geometric sequence will be negative if:

**the first term a**_{0 }is negative**the common ratio r is positive**

Of course, a geometric sequence may have some terms or no terms negative.

For example, if the common ratio r is negative, then every other term will be negative (and every other term will be positive).

If the first term a_{0} is positive and the common ratio r is positive, then all terms will be positive.

#### Example of A Geometric Sequence That Is Negative (All Terms Negative)

Consider the geometric sequence with first term a_{1} = -5 and common ratio d = 2. All of the terms in this sequence will be negative, since:

**the first term a0is negative****the common ratio r is positive**

We can prove this with the general formula for the nth term of a geometric sequence:

**a**[formula for nth term of any geometric sequence]_{n}= a_{0}r^{n}**a**[a_{n}= (-5)*2^{n}_{0}= -5, r = 2]**a**_{n}= -(5*2^{n})

Since 5*2^{n} is positive for any natural number n, we know that a_{n} = -(5*2^{n}) will be negative for any natural number n. Thus, every term of this geometric sequence will be negative.

The first several terms will be -5, -10, -20, -40, -80, -160, -320, -640, -1280, …

### Can Geometric Sequences Have Decimals Or Fractions?

Geometric sequences can have decimals or fractions. This can happen if either:

**the first term a**_{0 }involves decimals or fractions (a_{0}is not a whole number)**the common ratio r involves decimals or fractions (r is not a whole number)**

#### Example of A Geometric Sequence With Decimals

Consider the geometric sequence with first term a_{0} = 4 and common ratio r = 1/2. All but the first 3 terms in this sequence will have decimals, since:

**the common ratio r is not a whole number**

The first several terms will be 4, 2, 1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, …

### Is Compound Interest A Geometric Sequence?

You can think of compound interest as a geometric sequence. If the interest rate is R_{int}, then the common ratio is r = 1 + R_{int}.

For example, with an annual compound interest rate of 5% (R_{int} = 0.05), we have a geometric sequence with a common ratio of r = 1 + R_{int} = 1 + 0.05 = 1.05.

If we start with $10,000 (a_{0} = 10,000), we can calculate the balance after n years with the equation:

**a**[general formula for a geometric sequence]_{n}= a_{0}r^{n}**a**[with a_{n}= 10,000*(1.05)^{n}_{0}= $10,000 and r = 1.05]

For example, we can find:

**a**_{1}= 10,000*(1.05)^{1}**= $10,500**[the account balance after 1 year]**a**_{2}= 10,000*(1.05)^{2}**= $11,025**[the account balance after 2 years]**a**_{3}= 10,000*(1.05)^{3}**= $11,576.25**[the account balance after 3 years]

Note that simple interest is an arithmetic sequence.

You can learn more about the differences between simple interest and compound interest here.

### Is The Fibonacci Sequence A Geometric Sequence?

The Fibonacci Sequence is not a geometric sequence, since it does not have the proper form:

To find the nth term of the Fibonacci sequence, we use the formula:

**a**_{n}= a_{n – 1}+ a_{n – 2}

where a_{1} = 1 and a_{2} = 1.

Remember the key difference:

**Geometric Sequence: to get the next term, take the previous term and multiply by a constant value (the common ratio r, which never changes)****Fibonacci Sequence: to get the next term, take the previous term and add the 2**^{nd}previous term (the terms vary, and there is no common ratio between consecutive terms).

The first several terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, …

You can learn more about the Fibonacci sequence here.

## Conclusion

Now you know what geometric sequences are and what they look like. You also know the answers to some common questions about geometric sequences.

When you add up the terms of a geometric sequence, you get a geometric series, which you can learn more about in my article here.

You can learn more about increasing and decreasing sequences (and when they converge) here.

You might also want to read my article on arithmetic sequences.

I hope you found this article helpful. If so, please share it with someone who can use the information.

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~Jonathon