Arithmetic Sequences 📘
students, imagine you are saving money every week, and the amount you add stays the same each time. Or picture the seats in a theater where each row has the same number more seats than the row before it. These are examples of arithmetic sequences, a key idea in number and algebra. In this lesson, you will learn what arithmetic sequences are, how to describe them clearly, how to find any term in the sequence, and how they connect to real-life modelling in IB Mathematics: Applications and Interpretation SL.
What is an arithmetic sequence?
An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. That constant change is called the common difference, written as $d$.
For example, the sequence $3, 7, 11, 15, 19, \dots$ is arithmetic because each term increases by $4$, so $d=4$. Another example is $20, 17, 14, 11, \dots$, where each term decreases by $3$, so $d=-3$.
The terms of an arithmetic sequence are often written as $u_1, u_2, u_3, \dots$, where $u_n$ means the $n$th term. The first term is written as $u_1=a$, and the common difference is $d$.
The general formula for an arithmetic sequence is:
$$u_n=a+(n-1)d$$
This formula is one of the most important tools in this topic. It lets you find any term without writing out all the terms before it. For example, if $a=5$ and $d=3$, then:
$$u_n=5+(n-1)3$$
To find the $10$th term, substitute $n=10$:
$$u_{10}=5+9\cdot 3=32$$
This means the $10$th term is $32$.
Arithmetic sequences are part of number systems and numerical modelling because they help describe situations where a quantity changes by a fixed amount each time. That makes them useful in finance, science, and planning 📊.
How to identify and describe an arithmetic sequence
To check whether a sequence is arithmetic, subtract one term from the next term and see whether the difference stays the same.
Consider the sequence $8, 13, 18, 23, 28, \dots$.
The differences are:
$$13-8=5$$
$$18-13=5$$
$$23-18=5$$
$$28-23=5$$
Since the difference is constant, the sequence is arithmetic and $d=5$.
Now consider $2, 6, 12, 20, \dots$.
The differences are $4$, $6$, and $8$, which are not equal. So this is not an arithmetic sequence.
When describing an arithmetic sequence, always identify:
- the first term $a$
- the common difference $d$
- the general term formula $u_n=a+(n-1)d$
This language matters because IB mathematics expects you to explain mathematical ideas clearly, not just calculate answers. For example, you might write: “The sequence is arithmetic because the difference between consecutive terms is constant.”
A useful real-world example is monthly savings. Suppose students saves $15$ each week into an account, starting with $40$. The balance after each week forms an arithmetic sequence:
$$40, 55, 70, 85, \dots$$
Here, $a=40$ and $d=15$.
Finding terms and solving problems
The general term formula helps answer many types of questions.
Example 1: Find a specific term
Suppose $a=12$ and $d=4$. Find $u_8$.
Use the formula:
$$u_n=a+(n-1)d$$
Substitute the values:
$$u_8=12+(8-1)4$$
$$u_8=12+28=40$$
So the eighth term is $40$.
Example 2: Find the first term
Suppose $u_5=22$ and $d=3$. Find $a$.
Use the formula:
$$u_5=a+(5-1)3$$
$$22=a+12$$
$$a=10$$
So the first term is $10$.
Example 3: Find the common difference
Suppose $u_1=7$ and $u_6=27$. Find $d$.
Use:
$$u_6=a+(6-1)d$$
Since $a=7$:
$$27=7+5d$$
$$20=5d$$
$$d=4$$
So the common difference is $4$.
These steps show how arithmetic sequences are not just about memorizing formulas. They are about reasoning from given information and using algebra to find unknown values. This is a strong connection to Number and Algebra because the sequence rule itself is an algebraic expression.
The graph of an arithmetic sequence
Arithmetic sequences can also be shown on a graph. If you plot the term number $n$ on the horizontal axis and the term value $u_n$ on the vertical axis, the points lie on a straight line.
Why? Because the formula $u_n=a+(n-1)d$ can be rearranged into a linear form:
$$u_n=dn+(a-d)$$
This is a linear relationship between $u_n$ and $n$.
For example, if $a=3$ and $d=2$, then:
$$u_n=3+(n-1)2$$
$$u_n=2n+1$$
The points $(1,3)$, $(2,5)$, $(3,7)$, $(4,9)$, and so on lie on a straight line.
This graphing idea is useful in IB because technology can help you model and interpret patterns. Using a spreadsheet or graphing calculator, you can enter term numbers and values, then identify whether a pattern is linear. If the points form a straight line, that suggests an arithmetic sequence.
A key point is that sequence graphs use discrete points, not a continuous line. The points represent specific terms, so you should plot only whole-number values of $n$ unless a different context is given.
Arithmetic sequences in financial models
Arithmetic sequences often appear in situations where the change is constant. One common example is a fixed weekly allowance, a savings plan with equal deposits, or a payment plan with equal increases.
Imagine students earns $200$ in the first week of a part-time job and then earns $15$ more each week than the week before. The weekly earnings form an arithmetic sequence:
$$200, 215, 230, 245, \dots$$
Here, $a=200$ and $d=15$.
To find the earnings in week $12$:
$$u_{12}=200+(12-1)15$$
$$u_{12}=200+165=365$$
So the week $12$ earning is $365$.
In finance, arithmetic sequences can model regular increases, but they do not model growth by a fixed percentage. That means they are different from geometric sequences. For example, a salary that increases by $500$ each year is arithmetic, while an investment that grows by $5\%$ each year is geometric.
Understanding this difference helps with numerical modelling, another important part of IB Mathematics: Applications and Interpretation SL. The correct model depends on whether the change is additive or multiplicative.
Common mistakes and how to avoid them
Students sometimes confuse the common difference with the terms themselves. Remember: the common difference is the amount added or subtracted each time, not the terms of the sequence.
Another mistake is writing the formula incorrectly. The formula is:
$$u_n=a+(n-1)d$$
It is not:
$$u_n=a+nd$$
if $a$ is the first term, because the first term must occur when $n=1$.
Check this by substituting $n=1$:
$$u_1=a+(1-1)d=a$$
That works correctly.
A third mistake is forgetting that negative $d$ values are allowed. If the sequence decreases, such as $50, 45, 40, 35, \dots$, then $d=-5$.
A final useful habit is to label clearly. When solving a problem, write the values of $a$, $d$, and $n$ before calculating. This makes your reasoning easier to follow and reduces errors.
Conclusion
Arithmetic sequences are an important part of Number and Algebra because they combine pattern recognition, algebraic expressions, and real-world modelling. students, you should now be able to explain what makes a sequence arithmetic, use the common difference, apply the formula $u_n=a+(n-1)d$, and interpret arithmetic sequences in contexts such as savings, earnings, and linear growth. These skills build a strong foundation for later work with sequences, functions, and financial models in IB Mathematics: Applications and Interpretation SL.
Study Notes
- An arithmetic sequence is a sequence with a constant difference between consecutive terms.
- The constant difference is called the common difference and is written as $d$.
- If the first term is $a$, the $n$th term is given by $u_n=a+(n-1)d$.
- Arithmetic sequences can increase, decrease, or stay constant.
- To test whether a sequence is arithmetic, subtract consecutive terms and check whether the difference is constant.
- The graph of an arithmetic sequence is a set of discrete points on a straight line.
- Arithmetic sequences model situations with fixed additive change, such as weekly savings or fixed yearly increases.
- They are different from geometric sequences, which involve multiplication by a constant factor.
- In IB Mathematics: Applications and Interpretation SL, arithmetic sequences connect number patterns, algebraic reasoning, and numerical modelling.