An arithmetic sequence is a list of numbers that follow a definitive pattern. Each term in an arithmetic sequence is added or subtracted from the previous term.
For example, in the sequence \(10,13,16,19…\) three is added to each previous term. This consistent value of change is referred to as the common difference. A sequence can be increasing or decreasing, so the common difference can be positive or negative. For example, in the sequence \(90,80,70…\) the common difference is \(-10\).
Real-World Scenario
Arithmetic sequences are found in many real-world scenarios, so it is useful to have an understanding of the topic.
For example, if you earn \($55{,}000\) for your first year as a teacher, and you receive a \($2{,}000\) raise each year, you can use an arithmetic sequence to determine how much you will make in your \(12^{th}\) year of teaching. One way to solve this problem would be to write out each individual term in the sequence. The first number would be \($55{,}000\), the second number would be \($57{,}000\), and so on. If we did this \(12\) times, we would finally arrive at the number \($77{,}000\) for the \(12^{th}\) year of teaching.
Nth Term
The process of determining each term in a sequence can be very time-consuming, and rarely realistic. Fortunately, there is a quicker way to determine what is called the nth term, or any term in an arithmetic sequence.
The more efficient way solve for the \(n^{th}\) term in an arithmetic sequence is to use the formula \(a_n=a_1+(n-1)d\), where an represents the value of nth term, \(a_1\) represents the first term in the sequence, \(n\) represents the number of the term, and \(d\) represents the common difference. This formula allows us to quickly identify the value of any \(n^{th}\) term in an arithmetic sequence. In the teaching example mentioned previously, we can simply plug in the provided values into the formula in order to solve for the salary of the \(12^{th}\) year. The formula \(a_n=a_1+(n-1)d\) would become \(a_{12}=55{,}000+(12-1)(2{,}000)\), which simplifies to \(a_{12}=$77{,}000\).
Formula in a Decreasing Sequence
Now let’s look at an example of the formula being used in a decreasing sequence.
For example, Aimee is a professional scuba diver, and she descends \(6\) feet per minute. How deep will she be after \(17\) minutes if she starts at sea level? The same formula can be applied for decreasing sequences. When the relevant information is plugged into the formula, \(a_n=a_1+(n-1)d\) becomes \(a_{17}=0+(17-1)(-6)\), which simplifies to \(a_{17}=-96\), or \(96\) feet below sea level.
As you can see, the arithmetic formula for calculating the \(n^{th}\) term is a lot more efficient than taking the time to list out each term. It is important to remember that the formula \(a=-a_1+(n-1)d\) will only apply to sequences where the common difference is the result of addition or subtraction. Sequences that are built from multiplying or dividing each previous term are referred to as geometric sequences, and a different formula is used.
Writing Formulas for Arithmetic Sequences Sample Questions
Here are a few sample questions going over writing formulas for arithmetic sequences.
Which option shows the general formula for the \(n^{th}\) term in an arithmetic sequence?
The formula for the nth term in an arithmetic sequence is \(a_n=a_1+(n-1)d\). This formula can be used to determine the value of any term in an arithmetic sequence. An arithmetic sequence has a common difference between every term. For example: \(2 5,8,11…\) (adding \(3\) each time).
\(a_n=n^{th}\) term of the sequence
\(a_1=\) first term of the sequence
\(n=\) number of the term
\(d=\) common difference
Find the \(16^{th}\) term of the following arithmetic sequence.
\(2,10,18,26…\)
The original formula for the nth term of an arithmetic sequence is \(a_n=a_1+(n-1)d\). When the given values are plugged into the formula it becomes \(a_n=2+(16-1)(8)\). When simplified this becomes \(a_{16}=122\) which means that the \(16^{th}\) term in the sequence is \(122\).
An arithmetic sequence is a sequence whose terms increase or decrease by a _________.
common difference
common addition
common prime
common denominator
In an arithmetic sequence, the common difference is the amount that each term increases or decreases by. This consistent amount of change is the result of the same amount being added or subtracted from each term to the next. For example: \(3,6,9,12…\) or \(26,20,14,8…\)
Ryan is running a marathon next summer, so he wants to start a new training routine. He plans on running for \(8\) minutes this week, and then adding \(10\) minutes each of the following weeks. How many minutes will he run in the \(13^{th}\) week?
\(129\) minutes
\(128\) minutes
\(127\) minutes
\(126\) minutes
This problem can be solved using the formula for the \(n^{th}\) term of an arithmetic sequence. The formula \(a_n=a_1+(n-1)d\) becomes \(a_n=8+(13-1)(10)\), which simplifies to \(a_{13}=128\). This means that on the \(13^{th}\) week Ryan will be running for \(128\) minutes.
A poplar tree that measures \(2\) feet tall is planted in a park. The tree grows \(4\) inches each month. How tall will the tree be in \(8\) years?
\(604\) inches tall
\(484\) inches tall
\(554\) inches tall
\(404\) inches tall
The formula for the \(n^{th}\) term of an arithmetic sequence can be used to solve this problem. \(a_1\) can be found by converting \(2\) feet to inches by multiplying \(2\) by \(12\), which is equal to \(24\) inches. \(n\) can be found by finding the total number of months in \(8\) years: \(8\times12=96\text{ months}\). The formula \(a_n=a_1+(n-1)d\) will become \(a_n=24+(96-1)(4)\) when the values are plugged in. When this is simplified, it becomes \(a_{96}=404\), so after \(96\) months, or \(8\) years, the tree will be \(404\) inches tall.
