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 in 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 in the sequence (5 in the example above), and represents the value of the fifth term in the sequence (405 in the example above).
The Common Ratio
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 minus 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 one or more previous terms)
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 1 produces the starting value (100). Multiplying 100 by 0.1 produces the amount of growth (10). Adding the two values produces the new total (110).
Note that this is consistent with the idea that multiplying by a number between 0 and 1 (such as 10%) produces a result smaller than the starting value, while multiplying by a number larger than 1 (such as 1.1) will produce a bigger result.
So, when problem gives you a “rate of growth” or “rate of decay”, if you multiply the rate by the starting value, you will only get the amount of change (10 in the example above). This must be added to (growth) or subtracted from (decay) the starting value to produce the ending value.
On the other hand, if you add 1 to the rate of change (expressed as a positive value for growth, or a negative value for decay), you end up with a factor that takes you directly to the ending value in one calculation. In the example above, distributing 100 over (1 + 0.1) produces both the starting value (100) and the amount of growth (10) that must then be added together. However multiplying 100 by the sum of the two numbers in parentheses (1.1) produces a result (110) with starting value and growth already combined.
The “growth rate” form of the equation for a Geometric Sequence is useful whenever a problem is asking about a “rate”. The rate can end up being either positive (growth) or negative (decay). If you encounter a geometric sequence such as:
where the common ratio (R) is less than 1 (0.8 in this case), don’t let the fact that R is less than one bother you… 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:
This form of the equation allows you to quickly read the rate of growth or decay, without having to stop and 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
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.
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?
– 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?