# Arithmetic Sequences and Arithmetic Series

### Arithmetic Sequences / Progressions

The terms “sequence” and “progression” are interchangeable. An “arithmetic sequence” is the same thing as an “arithmetic 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 an arithmetic sequence (also known as an arithmetic progression).

Pick a number, any number, and write it down.  For example: $5$

Now pick a second number, any number (I’ll choose 3), which we will call the common difference. Now add the common difference to the first number, then write their sum down to the right of the first number: $5,~8$

Now, continue adding the common difference to the sum and writing the result down… over, and over, and over: $5,~8,~11,~14,~17,~20,~23,~26,~29, ...$

By following this process, you have created an “Arithmetic Sequence”, a sequence of numbers that are all the same distance apart when graphed on a number line:

### Vocabulary and Notation

In the example above 5 is the first term, or starting term, of the sequence. To refer to the starting term of a sequence in a generic way that applies to any sequence, mathematicians use the notation $a_1$

This notation is read as “A sub one” and means: the 1st value of the sequence 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 $a_1$ represents the value of the first term of the sequence (5 in the example above), and $a_7$ represents the value of the seventh term of the sequence (23 in the example above).

### The Common Difference

Since all of the terms in an Arithmetic Sequence must be the same distance apart by definition (3 apart in the example above), the magnitude of this distance is given a formal name (the common difference) and is often referred to using the variable $d$ (for Difference). If you add this value to any term, you end up with the value of the next term.  It can be calculated by subtracting the previous term from any term: $(a_2 - a_1)$ or $(a_7 - a_6)$, etc.

Every Arithmetic Sequence has a common difference between consecutive terms.  Examples include: $1,~3,~5,~7,~9,~...$ $10,~9,~8~,7,~6,~...$ $0,~0.4,~0.8,~0.12,~...$

The common difference 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 difference between all consecutive terms in that Arithmetic Sequence.

### Algebraic Description Of An Arithmetic Sequence

The existence of a common difference ( $d$) allows us to calculate terms in a generic way: $a_2~=~a_1+d$ $a_3~=~a_2+d$ $~~~~etc.$

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 a subscript: $a_n~=~a_{n-1}+d$

This would be read as “A sub N is equal to A sub N-1 plus the common difference d”. If $a_n$ refers to the “Nth” term, then $a_{n-1}$ has a subscript that is one less than N, and therefore refers to the term that immediately precedes $a_n$. A more intuitive way of reading this equation is “Any term may be calculated by adding the common difference to the previous term”.

These insights allow a complete description of an Arithmetic Sequence to take a number of forms: $5,~8,~11,~14, ...$
Specifying the first three or four terms is enough to demonstrate the common difference $a_1~=~5,~~d~=~3$
Specify the first term and the common difference $a_1~=~5,~~a_n~=~a_{n-1}+3$
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) $a_n~=~5+(n-1)(3)$
Specify a rule (based on the term number) for calculating the “Nth” term
This is a “Closed Form” definition (you only need to know the term number)

Note that if a sequence starts with a 5 then increases by 3 from one term to the next, this situation can be modeled using a linear equation with 5 as its y-intercept and 3 as its slope (with the domain restriction that “n” must be a positive integer). The last equation above uses this linear model, and provides the fastest way to calculate the Nth term of the sequence. Generalizing this linear equation approach leads to a description that applies to any Arithmetic Sequence: $a_n~=~a_1+(n-1)d$

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

If you know the first term of a sequence ( $a_1$), how many common differences do you need to add to it to get to the second term of the sequence ( $a_2$)? Since you seek the very next term, only one difference is needed: $a_2~=~a_1+d$

How many common differences are needed to get from the first to the third term? $a_3~=~a_1+d+d$

Now generalize the situation based on these two examples. When the term numbers were one apart (2 – 1 = 1), one common difference was needed to get from one to the other.  When the term numbers were two apart (3 – 1 = 2), two common differences were needed to get from one to the other. We will always need as many common differences as the difference between the two term numbers. For the general case, to get from $a_1$ to $a_n$, what is the difference between the two term numbers? One less than the value of “n”, or “n – 1”.

Thus “n-1” values of “d” must be added to the value of $a_1$ in order to arrive at the Nth term.

### Solving Arithmetic Sequence Problems

How many possible “unknowns” does the equation for $a_n$ have? $a_n~=~a_1+(n-1)d$

Four: $a_n,~a_1,~n,~and~d$. Therefore problems involving Arithmetic Sequences typically ask one of four questions:

What is the value of the Nth term? (Calculate the value for $a_n$)

What is the value of first term? (Solve for $a_1$)

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

What is the common difference? (Solve for “d”)

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 $a_{12}=24$, you can conclude that when “n” is 12, $a_n$ is 24, so you know two of the three bits of information you would need to answer a question about this sequence. Most Arithmetic Sequence problems can by solved by:

1. Determining the values for three of the four unknowns in the equation for $a_n$
2. Substituting those values into the equation above
3. 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.

For example, suppose the only information that a problem provides are values for  the 10th and 15th terms. You can find “d” either by a) taking the difference between the two terms and dividing it by 5 (the number of common differences needed to get from the 10th to the 15th term), or b) treating the 10th term as $a_1$, and the 15th term as $a_6$, then using the equation for $a_n$ to find “d”

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

Arithmetic Sequences can be thought of as linear equations with their domains restricted to integers. So they can model any situation that includes a constant rate of change, but where the only inputs that make sense are integers. Examples include:

• Manufacturing situations, where the total quantity of finished product produced depends on the number of machine cycles completed
• Product pricing, where the total price equals a fixed amount for shipping and handling plus an amount per unit ordered
• Game scores for games with a fixed point value per goal scored

### Arithmetic Series

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

Consider the Arithmetic Sequence described at the beginning of this post: $5,~8,~11,~14,~...$

The series based on this sequence is: $5,~13,~24,~38,~...$

The 3rd term of the Series is the sum of the first three terms of  the underlying sequence, and is typically described using Sigma Notation with the formula for the Nth term of an Arithmetic Sequence (as derived above): $S_3=\displaystyle\sum_{i=1}^{3}(5+(i-1)3)\\*~\\*~\\*=5+8+11\\*~\\*=24$

### Formula for the Nth Term

Just it is sometimes useful to have a formula for the Nth term of an Arithmetic Sequence, it is also useful to have a formula for the Nth term of an Arithmetic Series, which avoids having to a add up a long list of terms.

Arithmetic Sequences have a useful pattern to them that leads to the formula we seek. Suppose we need to find the sum of the first six terms of this Arithmetic Sequence: $5,~8,~11,~14,~17,~20,~...$

The difference between successive terms in this sequence (“d”) is always 3 . What happens when, instead of adding the first two terms together, we add the pair of terms on outside first, then the next pair of “in” from them, etc.: $5+20=25\\*~\\*8+17=25\\*~\\*11+14=25$

### How Does This Pattern Lead To A Formula?

Three insights lead us to a formula for the sum of an Arithmetic Sequence.

1) Why do the above pairs always add to the same number? Think about the second pair that was added above: $(a_2+a_5)$. $a_2$ is 3 greater than $a_1$, while $a_5$ is 3 less than $a_6$. So $a_2+a_5\\*~\\*=(a_1+3)+(a_6-3)\\*~\\*=a_1+a_6+3-3\\*~\\*=a_1+a_6$

The two differences of three (one positive and one negative) cancel each other out, and the sum of the second pair will be the same as the sum of the first pair. This will be true for any Arithmetic Sequence (if you substitute “d” for “3” above).

2) What is the significance of the number that these number pairs add to (25 in the example above $=a_1+a_6$)? Consider what would happen if we were looking at the first seven (an odd number) terms of the sequence… there would be a term “left over” after we have paired off as many terms as we could, with each pair having the same sum as all the other pairs (28 in this case):

5, 8, 11, 14, 17, 20, 23

That remaining term will be the “middle” term in the sequence.  Both members of each number pair (for example, 5 and 23) will be equally distant from that middle term, one to the left and one to the right. That makes the middle term the average of each of these “outer” pairs. The middle term will also be the average of the first “n” terms: $\dfrac{(a_1+a_n)}{2}$ is the average of the first “n” terms

If there are an even number of terms, there won’t be a “middle” term but every pair will still have the same sum, and therefore the same average (equal to half their sum). Therefore the sum of each pair, the quantity that ended up always being 25 or 28 above, will always be twice the average of the first “n” terms in the sequence.

3) How can knowing the average of the first “n” terms help? Recall that the average of a set of terms is equal to the sum of all terms, divided by the total number of terms. If you know the average of a sequence, and you know the number of terms in the sequence, then you can easily calculate the sum of all terms: $Average=\dfrac{(Sum~of~Terms)}{(Number~of~Terms)}$

Multiplying both sides of this equation by the Number of Terms produces: $(Number~of~Terms)(Average)=(Sum~of~Terms)$

This provides an easy way to calculate the sum of the first “n” terms of an Arithmetic Sequence. Since we now know that

Average of first n terms $=\dfrac{(a_1+a_n)}{2}$

we can multiply it by “n” to arrive at a simple formula for the sum of the first “n” terms of any Arithmetic Sequence:

Sum of first n terms $=\displaystyle\sum_{i=1}^{n}a_i=n\cdot \dfrac{(a_1+a_n)}{2}$

Therefore, the formula that describes the Nth term of an Arithmetic Series (where the Nth term is the sum of the first N terms of the underlying Arithmetic Sequence) is: $S_n=n\cdot \dfrac{(a_1+a_n)}{2}$

Note that $a_n$ is usually used to represent the value of the Nth term of an Arithmetic Sequence, while $S_n$ (S for Sum) is usually used to represent the value of the Nth term of an Arithmetic Series.

### Solving Arithmetic Series Problems

Just as with Arithmetic Sequence problems, there are four possible unknowns in an Arithmetic Series problem: $S_n$ $n$, $a_1$, and $a_n$. So the four types of questions that are typically asked are:

What is the value of the Nth term? (Calculate the value for $S_n$)

What is the value of first term? (Solve for $a_1$)

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

What is the Nth term of the underlying Arithmetic Sequence? (Solve for $a_n$)

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.  From there, algebra skills should get you to an answer for the question.

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

An Arithmetic Sequence describes something that is periodically growing in a linear fashion (by the same amount each time), and an Arithmetic Series describes the sum of the periodic values.  Examples of Arithmetic Series include:

– The total number of seats in a fan-shaped auditorium, where each row has a two more seats than the previous one

– The total amount of grain produced over 10 years if farmers produce 50 tons more than the previous year every year

More things in our world seem to grow exponentially than linearly, so you will probably run into more applications of Geometric Series than Arithmetic Series. However, the mathematical approaches learned in working with Arithmetic Sequences and Series will serve as a good background for working with Geometric Sequences and Series. ### Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.

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