What Is A Geometric Sequence? (10 Common Questions Answered)

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 = arⁿ.

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_nr

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 = a₁ / a₀

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 = a₁₁ / a₁₀

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 a₁ = 8 and a₂ = 32, we would calculate:

d = a₂ / a₁

=32 / 8
=4

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

geometric sequence a0 = 1 r = 4 — The first several numbers in the geometric sequence with first term 2 and common ratio 4. Note that the curve displays exponential growth.

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 a₃ = 8 and a₁₂ = 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 = (a₁₂ / a₃)^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 a₀ if we want to:

a₃ = 8

a₂ = 4 [since a₃ = a₂r, with a₃ = 8 and r = 2]
a₁ = 2 [since a₂ = a₁r, with a₂ = 4 and r = 2]
a₀ = 1 [since a₁ = a₀r, with a₁ = 2 and r = 2]

We can find a general formula for the first term a₀ by solving the equation from before with m = 0:

r = (a_n / a_m)^1/(n-m)
r = (a_n / a₀)^1/(n-0) [use a = 0]

r = (a_n / a₀)^1/n
r = (a_n / a₀)^1/n
rⁿ = a_n / a₀ [raise both sides to the power of n]

rⁿ / a_n = 1 / a₀ [divide both sides by a_n]
a_n / rⁿ = a₀ [take the reciprocal of both sides]

So, given the nth term a_n and the common ratio r of a geometric sequence, we can find the first term a₀ by the equation a₀ = a_n / rⁿ.

Does A Geometric Sequence Always Increase?

A geometric sequence does not always increase. For example, if the first term a₀ is positive and the common ratio r is positive and less than 1, the geometric sequence will decrease.

Take a₀ = 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.

geometric sequence a0 = 8 r = 1 over 4 — The first several numbers in the geometric sequence with first term 8 and common ratio 1/4. Note that the curve displays exponential decay.

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 a₀ = 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.

geometric sequence a0 = 1 r = -2 — The first several numbers in the geometric sequence with first term 1 and common ratio -2. Note that the points oscillate between positive and negative values, increasing in absolute value.

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 rⁿ approaches zero when |r| < 1).

If r = 1, the geometric sequence converges to a₀ (a ratio of r = 1 means all the terms have the same value of a₀ and the sequence converges to a₀). The terms would be a₀, a₀, a₀, a₀, …

If r = -1, the geometric sequence oscillates between a₀ and –a₀. The terms would be a₀, -a_0,a₀, -a₀, …

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 a₀ and the common ratio r.

We can use the equation for a₀ we found above and solve for a_n instead of a₀:

a_n / rⁿ = a₀ [equation from above]
a_n = a₀rⁿ [multiply by r_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_nby the equation a_n = a₀rⁿ_.

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

Let’s say we are given the terms a₅ = 486 and a₈ = 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 a₀. We will use the formula for a₀ that we found earlier, along with a₅ = 486:

a₀ = a_n / rⁿ
a₀ = a₅ / r⁵[use n = 5]
a₀ = 486 / 3⁵[a₅ = 486 and r = 3]

a₀ = 486 / 243
a₀ = 2

Now we can find the formula for the nth term a_n by the equation from earlier:

a_n = a₀rⁿ
a_n = 2*3ⁿ [a₀ = 2, r = 3]

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

a₀ = 2*3⁰ = 2*1 = 2 (correct!)
a₅ = 2*3⁵ = 2*243 = 486 (correct!)
a₈ = 2*3⁸ = 2*6,561 = 13,122 (correct!)

So, the formula for the nth term of this geometric sequence is a_n = 2*3ⁿ.

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

geometric sequence a0 = 2 r = 3 — The first several numbers in the geometric sequence with first term 2 and common ratio 3. Note that the curve displays exponential growth.

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_nr

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 a₀is 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 a₀ 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 a₁ = -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_n = a₀rⁿ [formula for nth term of any geometric sequence]
a_n = (-5)*2ⁿ [a₀ = -5, r = 2]
a_n = -(5*2ⁿ)

Since 5*2ⁿ is positive for any natural number n, we know that a_n = -(5*2ⁿ) 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, …

geometric sequence a0 = -5 r = 2 — The first several numbers in the geometric sequence with first term -5 and common ratio 2. Note that the curve displays exponential growth, but in a negative direction.

Can Geometric Sequences Have Decimals Or Fractions?

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

the first term a₀involves decimals or fractions (a₀ 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 a₀ = 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, …

geometric sequence a0 = 4 r = 1 over 2 — The first several numbers in the geometric sequence with first term 4 and common ratio 1/2. Note that the curve displays exponential decay.

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 (a₀ = 10,000), we can calculate the balance after n years with the equation:

a_n = a₀rⁿ [general formula for a geometric sequence]

a_n = 10,000*(1.05)ⁿ [with a₀ = $10,000 and r = 1.05]

For example, we can find:

a₁ = 10,000*(1.05)¹ = $10,500 [the account balance after 1 year]

a₂ = 10,000*(1.05)² = $11,025 [the account balance after 2 years]
a₃ = 10,000*(1.05)³ = $11,576.25 [the account balance after 3 years]

geometric sequence a0 = 10000 r = 1.05 (compound interest) — This graph shows the first 40 years of compound interest at 5% for a starting balance of $10,000. Note that the curve displays exponential growth.

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 a₁ = 1 and a₂ = 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, …