### Geometric Sequences / Progressions

The terms “sequence” and “progression” are interchangeable. A “geometric sequence” is the same thing as a “geometric progression”. This post uses the term “sequence”… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create a geometric sequence (also known as a geometric progression).

Pick a number, any number, and write it down. For example:

Now pick a second number, any number (I’ll choose 3), which we will call the **common ratio**. Now **multiply **the first number by the common ratio, then write their product down to the right of the first number:

Now, continue multiplying each product by the common ratio (3 in my example) and writing the result down… over, and over, and over:

By following this process, you have created a “Geometric Sequence”, a sequence of numbers in which the ratio of every two successive terms is the same.

### Vocabulary and Notation

In the example above, 5 is the **first term** (also called the starting term) of the sequence or progression. To refer to the first term of a sequence in a generic way that applies to any sequence, mathematicians use the notation

This notation is read as “A sub one” and means: the 1st value of the sequence or progression represented by “a”. The one is a “subscript” (value written slightly below the line of text), and indicates the position of the term within the sequence. So represents the value of the first term of the sequence (5 in the example above), and represents the value of the fifth term of the sequence (405 in the example above).

Since all of the terms in a Geometric Sequence must be the same multiple of the term that precedes them (3 times the previous term in the example above), this factor is given a formal name (the **common ratio**) and is often referred to using the variable (for Ratio). If you multiply any term by this value, you end up with the value of the next term.

For an existing Geometric Sequence, the common ratio can be calculated by dividing any term by its preceding term:

Every Geometric Sequence has a common ratio between consecutive terms. Examples include:

The common ratio can be positive or negative. It can be a whole number, a fraction, or even an irrational number. No matter what value it has, it will be the ratio of **any** two consecutive terms in the Geometric Sequence.

Therefore, to test if a sequence of numbers is a Geometric Sequence, calculate the ratio of successive terms in various locations within the sequence. If you calculate the same ratio between any two adjacent terms chosen from the sequence (be sure to put the later term in the numerator, and the earlier term in the denominator), then the sequence is a Geometric Sequence. One of the series shown above can be used to demonstrate this process:

Since the ratio between adjacent terms was always equal to the same number (negative one third), this is a Geometric Sequence.

### Algebraic Description Of An Geometric Sequence

The existence of a common ratio allows us to calculate terms in a generic way:

Since every line above follows the same pattern, the whole process can be described a bit more generally and compactly by using a variable as the subscript:

This would be read as “A sub N is equal to A sub N-1 times the common ratio R”. If refers to the “Nth” term, then has a subscript that is one less than N, and therefore refers to the term that immediately precedes . A more intuitive way of reading this equation is “Any term may be calculated by multiplying its preceding term by the common ratio”.

These insights allow a complete description of a Geometric Sequence to take a number of forms:

Specifying the first three or four terms is enough to demonstrate the common ratio

Specify the first term and the common ratio

Specify the first term, with a rule to get you from each term to the next

This is a **recursive** definition (you must know the previous term)

Specify a rule (based on the term number) for calculating the “Nth” term

This is a **closed form**, or **explicit** definition (you only need to know the term number)

Note that if a sequence starts with a 5 then grows by a factor 3 from one term to the next, this situation can be modeled using an exponential equation with 5 as its initial value and 3 as its base (with the domain restriction that “n” must be a positive integer). The last equation above uses this exponential model, and provides the fastest way to calculate the Nth term of the sequence. Generalizing this exponential equation approach leads to a description that applies to any Geometric Sequence:

### Why Is (n-1) In The Equation?

If you know the first term of a sequence (), how many common ratios do you need to multiply it by to get to the second term of the sequence (? Since you seek the very next term, only one use of the common ratio is needed:

How many common ratios are needed to get from the first to the third term?

Now generalize the situation based on these two examples. When the term numbers were one apart (2 – 1 = 1), one use of the common ratio was needed to get from one to the other. When the term numbers were two apart (3 – 1 = 2), two uses of the common ratio were needed to get from one to the other. We will always need to use the common ratio as many times as the difference between the two term numbers. For the general case, to get from to , what is the difference between the two term numbers? One less than the value of “n”, or “n – 1”.

Thus the common ratio must be used “n-1” times to get from the value of the first term to the value of the n’th term.

### Another Version Of The Equation

While the equation above (and copied below) applies to any geometric sequence

there is a very similar version you will also see used often, which is the “growth rate” form of the equation:

Note that the only difference between the two is that the “R” in the first equation becomes “1+r” in the second. Why bother with two versions? Because the second is more convenient when working with problems involving exponential growth or decay, or interest rates. Suppose a geometric sequence is describing a 10% growth pattern:

what is the common ratio, R?

Why does the common ratio end up being 1.1 when the growth rate is 10% or 0.1? The distributive property of multiplication over addition helps explain this:

Multiplying 100 by 10% produces the amount of growth (10), but NOT the new total (110). This is consistent with the fact that multiplying by a number between 0 and 1 (such as 10%) will produce a **smaller** result, while multiplying by a number larger than 1 (such as 1.1) will produce a **bigger** result. So, multiplying 100 by (1 + r) or (1 + 0.1), produces the starting value (100) plus the amount of growth (10), and their sum (110) is 10% greater than the starting value (100).

If you encounter a geometric sequence such as:

the common ratio (R) is 0.8. To fit this into the “growth rate” form, solve the equation

so the growth rate in this problem must be a **negative** 20%, which implies that exponential decay is being modeled, and the “growth rate” form of the equation is:

Note how this form of the equation allows you to quickly read the rate of growth or decay, without having to stop and think to figure out what decay rate produces a common ratio of 0.8.

### Solving Geometric Sequence Problems

How many possible “unknowns” does either equation for have?

Four: . Therefore problems involving Geometric Sequences typically ask one of four questions:

What is the value of the Nth term (calculate the value of )?

What is the value of first term (solve for )?

Given a value, what term number must it be (solve for “n”)?

What is the common ratio (solve for “R”) or rate of growth/decay (solve for “r”)?

To answer one of the above questions, you must know (or be given enough information to determine) values for three of the “unknowns” in the equation above. For example, if you are told that , you can conclude that when “n” is 12, is 4,096, so you know two of the three bits of information you would need to answer a question about this sequence. Most Geometric Sequence problems can by solved by:

- Determining the values for three of the four unknowns in the equation for
- Substituting those values into the appropriate equation above
- Solving for the only variable remaining

Some problems will be a little more complex, but you should still be able to use the information provided to determine values for three of the four unknowns. Note that problems which ask you to solve for the term number, which is in the exponent, may require the use of logarithms during the solution process.

For example, suppose the only information that a problem provides are values for the 10th and 15th terms. You can find “R” either by a) taking the ratio of the two terms, then taking its 5th root (the number of common ratios needed as factors to get from the 10th to the 15th term), or b) treating the 10th term as , and the 15th term as (their term numbers are 5 apart, just as 10 and 15 are), then using the equation for to find “R”.

### Applications of Geometric Sequences in “Real Life”

Geometric Sequences can be thought of as exponential equations with their domains restricted to integers. So they can model situations that involve a constant rate of growth, but where the only inputs that make sense are integers. Examples include:

– Annual size of a population that is growing (or shrinking) at a constant rate

– Value of money in an account that receives periodic fixed rate interest payments

– Maximum height of a bouncing ball after each bounce (when a fixed % of its energy is lost on each bounce)

– Radioactivity levels of a sample over time

### Geometric Series

The Nth term of a “series” is the **sum** of the first N terms of its underlying sequence.

Consider the Geometric Sequence described at the beginning of this post:

The **series** based on this **sequence** is:

The 3rd term of the Series (65) is the sum of the first three terms of the underlying sequence (5 + 15 + 45), and is typically described using Sigma Notation with the formula for the Nth term of an Geometric Sequence (as derived above):

### Formula for the Nth Term

Just as it is sometimes useful to have a formula for the Nth term of an Geometric **Sequence**, it is also useful to have a formula for the Nth term of an Geometric **Series**, which allows us to avoid having to a add up a long list of terms.

The fact that Geometric Sequences are based on multiplication creates a useful pattern which leads to the formula we seek. Suppose we wish to find the sum of the first N terms of a Geometric Sequence. Let’s express each term using the formula above, so that it involves only and “R”, and show only the first and last two terms:

Note that in the Nth term, “R” has an exponent of “n-1”, not “n”. If we multiply both sides of the above equation by “R”, we end up with:

Now arrange this equation next to the first one, so that we are ready to create a linear combination of the two to solve for , as you have hopefully done when solving systems of linear equations. Lining up like terms above one another, what do you notice?

Hmmm… all but two of the like terms on the right side of the equation are exactly the same as one another. So, if we proceed with solving these two equations as a system, and subtract one from the other, then solve for , we get:

This formula will produce the Nth term of a Geometric **Series**, which will equal the sum of the first N terms of its underlying Geometric **Sequence**.

### Infinite Geometric Series

What happens when we take the sum of an infinite number of terms in a sequence?

Let’s explore this by first considering Arithmetic (**not** Geometric) Sequences. As the number of terms in an Arithmetic Sequence grows, the last term moves farther and farther from zero as the common difference is added repeatedly. So the last term in an Arithmetic Sequence will **always** approach infinity (either positive infinity, or negative infinity) as the number of terms approaches infinity, except when the common difference is zero.

If an Arithmetic Sequence’s last term approaches infinity, or even if it is constant (due to a common difference of zero) then the sum of all those terms must approach infinity. Therefore, **all** Arithmetic **Series** (except 0, 0, 0, …) will approach infinity as N grows large.

Turning to Geometric Sequences, consider what happens to an exponential expression when:

a) the absolute value of “R” is greater than 1

b) the absolute value of “R” is between 0 and 1

When a value greater than 1 is raised to a positive power, it grows… the larger the base, the faster it grows:

So, when the common ratio is greater than 1, the Nth term of a Geometric Sequence will also grow towards infinity as N gets larger. Therefore a Geometric Series with an “R” greater than 1 will grow towards infinity with each additional term.

However, when the absolute value of the common ratio is between -1 and 1, a special situation arises. What happens when a value between zero and one is raised to a power? It shrinks… the closer to zero it is, the faster it shrinks towards zero:

Using this insight, and re-examining the formula for the Nth term of a Geometric Sequence:

you can conclude that if the value of “R” is between -1 and 1, then the value of Nth term must move closer and closer to zero as N grows. So, the sum of all those terms could perhaps get closer and closer to some limit, which would represent the exact sum of an infinite number of terms.

Now let’s re-examine the formula for the Nth term of a Geometric Series with this in mind:

If the common ratio is a value between -1 and 1, as “n” gets infinitely large must become infinitely close to zero. Substituting zero for in the formula above, and dropping the “n” subscript on the initial “S” because we are considering the sum of **all** terms, produces:

So “S” is the value that the Nth term of the Geometric Series approaches as N becomes infinitely large, which is equal to the sum of all (an infinite number of) terms in the underlying geometric sequence.

### An Example

This formula allows us to easily find the sum of the infinite Geometric Sequence

by first determining that , , then using these values in the above formula

Therefore, the sum of this infinite Geometric Sequence is the integer 4. Did you expect that an infinite sequence of increasingly small fractions would sum to such a round number?

### Solving Geometric Series Problems

Just as with Geometric Sequence problems, there are up to four possible unknowns in an Geometric Series problem: . If the problem involves an infinite series, there are three unknowns. So the four types of questions that are typically asked are:

What is the of the Nth term of the Series? (Calculate the value of )

What is the value of first term? (Solve for )

What is the common ratio? (Solve for )

Given a value, what term number must it be? (Solve for “n”… may require using logarithms)

Or, for problems involving infinite Geometric Series:

What is the sum of all terms in the infinite sequence? (Solve for “S”)

To answer one of the above questions, you must know (or be given enough information to determine) values for all but one of the “unknowns” in the equations above. From there, algebra skills should get you to an answer for the question.

### Applications of Geometric Series in “Real Life”

An Geometric **Sequence** describes something that is periodically growing in an exponential fashion (by the same percentage each time), and a Geometric **Series** describes the sum of those periodic values. Examples of Geometric Series that could be encountered in the “real world” include:

– What is the total number of births over a 20 year period to a population that grows at a fixed percentage each year?

– How much interest will $1,000 invested in a fixed-rate certificate of deposit earn over 30 years?

– How much of a medicine that a patient takes every 8 hours remains in the body after taking it for 48 hours if only 20% remains after each 8 hour period?

– If consumption of a rare material is growing at 5% per year, how much will be consumed in total over the next 10 years?

– Does

– If a ball that is dropped bounces back to 80% of its previous height after each bounce, how far up and down will it have travelled after 15 bounces?

Thanks for this thorough examination of the topic of geometric series. I shared this with two of my colleagues this morning.

Thank you! Suggestions, questions, comments are always welcome.

One of my favorite problems that tie together this and the arithmetic series post is – Insert two numbers between 4 and 24 so that the first three numbers form a geometric progression while the last three numbers form an arithmetic progression

Ooooh – I like it! Good algebra problem that feels like it will get a little messy, but tie together problem formulation,sequence, and system skills.

Mr. Whitford may i ask what type of series is this sequence of numbers 1, 7, 19, 37, 61, 91, 127, …if you add the first and second terms youll get 2^3 plus another term 3^3 and so on. Also, are they figurate numbers? Is a study about this before? Thanks you very much.

There are many, many sequences which are neither arithmetic nor geometric. Some may be able to be described with an explicit (or closed firm) formula, others may only be described with a recursive definition, and some may defy attempts to describe them with a formula.

I regret that your question goes beyond my experience, as I have spent most of my time with arithmetic and geometric sequences, and yours is neither. Perhaps another reader of this page can provide you with an answer.

Please help me calculating the average growth rate of this series:

1.5 1.6 1.3 1.1 -0.08 3.6 -0.19

Thank you!

I don’t think I have ever seen a question quite like this… The series is neither arithmetic, nor geometric, but the question asks for an “average growth rate”, not an “average difference”, so I assume we are meant to treat it as though it were a geometric sequence.

Two approaches come to mind:

1) A statistical one: Look at the ratio of every pair of successive terms, and average that ratio. That seems to follow the instructions, but will not necessarily get you from the first term to the last term exactly.

2) A goal oriented one: Divide the last term by the first term, take the 6th root of that ratio, then subtract one to arrive at a “growth rate” (R) that will get you from the first term to the last term after applying it six times: 1.5 * (1+R)^6 = -0.19

Given that I am not really certain about the goal of the question (to calculate the last term exactly, or to describe in an “average” way the change that happens from one term to the next), I am not certain which of the approaches above is being asked for.

I am a retired missionary, and the numbers are the average quinquennial growth rates of church membership from 1980 – 2015 of a certain region of our denomination. There are two that are negative. I would like to know the average % with which this series changes. Does this help?

Yes, that helps a lot! I did not realize that the numbers were already growth rates… I thought they were (in effect) the membership numbers.

If the numbers are already growth rates, you can still take the same two approaches… but the calculations will be a little different.

Your average growth rate is (1.5 + 1.6 + 1.3 + 1.1 – 0.08 + 3.6 – 0.19) / 7 = 1.26% This is a number that is easy to calculate and understand. It describes the average growth rate of the congregation over the past 8 periods. However, if you apply that growth rate to the initial membership 7 times, it is not likely to produce the

exactfinal membership number.If you want the exact mathematical “average” growth rate, you can find it be calculating: [(Membership in in period 8) / (Membership in period 1)] ^ (1/7)

then subtract 1from the result to convert the “periodic growth factor” into a “periodic growth rate” = R. This is the number that will produce an exact result:(Membership in period 1)*(1+R)^(7)=(Membership in period 8). Based on the percentages in your comment, it should end up being 1.2547%

Thank you so much. Could you give me the sequence for a scientific calculator so that I can do this? Thanks.

I showed the steps used to calculate both numbers if you have the membership totals.

The only thing I did not show was calculating the exact 8-period growth rate from the 7 individual ones. To do that

1) convert each percentage into a decimal and add one to it:

1.5 becomes 1 + 0.015 = 1.015

-.08 becomes 1 – 0.0008 = 0.9992

These numbers are the growth factors for each period.

2) multiply all 7 of the values you calculated in step 1 to produce a single number. This number is the 8-period total growth factor. It will be the same number you get by calculating the ratio: (Membership in period 8) / (Membership in period 1)

3) Since there are seven growth periods between the first and last membership period, take the 7th root of the result of step 2. On a scientific calculator, you generally do that by raising the result to the 1/7 power: R ^ (1/7). This number is the overall quinquennial growth factor, which when applied 7 times to the (Membership in period 1), will produce (Membership in period 8).

4) To convert the result of step 3 into a percentage growth rate, subtract one and multiply by 100. This will be the (average)

quinquennialpercentage growth rate from 1980 to 2015.5) To calculate an

annualpercentage growth rate, take the 35th root of the result of step 2 (since there are 5 years in each period, times 7 periods), subtract 1, and multiply by 100.Your help is very much appreciated. I was able to calculate the average annual growth (geometrical mean) rate with positive numbers but not with negative ones. Best wishes!

You are most welcome.

Many people neglect to turn rates of growth or decay into their corresponding factors before using them.

(Starting balance)(growth/decay rate) = (amount of change)

This is not a big deal if only applying them over one period, as the (amount of change) can easily be added to the (Starting balance) to arrive at an (Ending balance):

(Ending balance) = (Starting Balance) + (Starting Balance)(growth/decay rate)

However, when making multi-period calculations, the above approach becomes very time consuming. If you remember some of your high school algebra, you might notice a common factor on the right side above, which can be factored out:

(Ending balance) = (Starting Balance)[1 + (growth/decay rate)]

The amount in square brackets is what I refer to as the “growth factor”, and is the quantity that gets you directly to an ending balance. This allows the growth factors for successive periods to all be (relatively easily) multiplied together to arrive at a single “net” growth factor that describes the growth across all of the periods:

(Starting Balance) * [1 + (period 1 growth/decay rate)] = (Per 1 Ending balance)

(Per 1 Ending Balance) * [1 + (period 2 growth/decay rate)] = (Per 2 Ending balance)

or

(Starting Balance) * [1 + (period 1 growth/decay rate)] * [1 + (period 2 growth/decay rate)] = (Per 2 Ending balance)

so

[1 + (period 1 growth/decay rate)] * [1 + (period 2 growth/decay rate)] = (net growth factor over Periods 1 and 2)

By taking the appropriate root of the “net” growth factor, then subtracting one and expressing it as a percentage, you arrive at an “average growth/decay rate”.

Thank you for this article. You touch on using logarithms in the solutions of geometric sequences. I have a question as follows:

Find how many terms there are in this geometric sequence:

-1, 2, -4, 8, …….., -16 777 216

I am trying to use logs to solve it (the question is in a logs exercise) and I end up needing to find the log of -2 (the common ratio), which of course is not possible. I get the correct answer when I use +2 instead, and I have seen elsewhere a comment aluding to the use of the modulus value, but I have no explanation as to why this would be considered acceptable.

Please could you help me by explaining? Many thanks.

Helen,

This problem can be solved by considering sign and magnitude separately. Since the signs of the first and last terms are negative, there must be an odd number of negative factors in the last term (N must be odd, while the base is -2 as you said). Or to use the term “modulus”, the exponent mod 2 must equal 1 (there must be a remainder of 1 when the exponent is divided by 2).

Now we turn to considering the magnitude, and to do that we ignore the sign of the base (for the reason you mention).

So, after using the base change formula to evaluate the log on a calculator, you will need to verify that your solution is indeed odd. If it is not odd, then it violates the requirement identified in the first paragraph, which would mean that there is no solution to this problem.

Does that answer your question?

Thank you for such a quick response, Mr Ford.

I understood what you said, apart from ‘now we turn to considering the magnitude, and to do that we ignore the sign of the base…..’

Why can we do that? In every other bit of maths we cannot ignore a negative sign, so why can we do that here? (I apologise if I am missing something obvious here).

Well, we are not ignoring the sign of the base entirely, just analyzing sign and magnitude in separate steps. Some problems cannot be solved in a completely sequential fashion.

For example, when solving a quadratic equation in factored form that is equal to zero, we break the problem up into two pieces (using the zero product property), and solve each piece separately. This problem is similar, in that the mathematical tool we wish to use (logarithms) will not work when the base is negative.

By using your understanding of both patterns at work in the problem (sign, magnitude), and analyzing each separately, you can find, and defend, a solution.

Thank you very much, sir. I will need to have a think about it as my understanding is not up to yours. But I will get there. Your help is much appreciated, thank you.

Hello, what if the ratio is a percentage but the terms must be bigger than the previous term, for example 2% bigger?

This is a problem I have encountered and I cannot answer because the answer i get is either way too small & far or way too big and also far from the answer.

-A town with a current population of 150, 000 citizens is growing at the rate of 2% per year. Find the population of the town 20 years from now.

In your example, they are not giving you the “common ratio”, but instead are giving you the “growth rate”. So, you can use either the form of the common ratio, where “r” is the rate of growth (0.02) or the form where “R” is the common ratio (1.02 in your example). So, your example describes a geometric sequence where

r = 0.2 or R = 1.02

n = 20

= 150,000

and they are asking for

Thank you for the answer.

If A sub n = 150,000×1.02^19

A sub n = 218,521.68 (rounded off) is this correct?

But the formula i was taught is a sub 1 × r^n-1 so if i followed that…

150000×0.2^19

= 7.86432×10^-09 on the calcu…

By the way, is r & R different?

Thank you very much in advance.

Your first calculation is correct, your second is not. R and r are different. As used in my blog post above, but applied to your question, R = 1.02 while r = 0.02. “R” is the “common ratio” of a geometric sequence, while “r” is the growth or decay rate in the problem… which must have a 1 added to it to become the common ratio of a Geometric Sequence.

Ahh, I could see now. I have read your rule about ratios < 1 and the growth rates. So, to conclude what I have learned, you would have to add 1 to the r if r is between 1 & 0 in order for it to grow?

If you are given a “growth rate” or a “decay rate”, you must add 1 to it before raising it to the power of the number of time periods.

If you are given a “common ratio”, do NOT add 1 to it before raising it to a power.

If the growth rate is 257%, the common ratio will be 1 + 2.57 = 3.57

If the rate of decay is 35%, the common ratio will be 1 – 0.35 = 0.65

If the common ratio is 1.25, the growth rate is 1.25 – 1 = 0.25 or 25%

If the common ratio is 0.78, the growth rate is 0.78 – 1 = -0.22 or -22% (actually decay, not growth)