Writing Formulas for Arithmetic Sequences

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 $10,13,16,19\dots$ 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 $90,80,70\dots$ 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 $\$\mathrm{55,000}$ for your first year as a teacher, and you receive a $\$\mathrm{2,000}$ 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 $\$\mathrm{55,000}$, the second number would be $\$\mathrm{57,000}$, and so on. If we did this $12$ times, we would finally arrive at the number $\$\mathrm{77,000}$ 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 $a_n$ 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 $a_{12}=\mathrm{55,000}+(12-1)(\mathrm{2,000})$, which simplifies to $a_{12}=\$\mathrm{77,000}$.

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.