# Explicit formulas for arithmetic sequences

CCSS Math: HSF.LE.A.2
Learn how to find explicit formulas for arithmetic sequences. For example, find an explicit formula for 3, 5, 7,...
Before taking this lesson, make sure you are familiar with the basics of arithmetic sequence formulas.

# How explicit formulas work

Here is an explicit formula of the sequence $3, 5, 7,...$
$a(n)=3+2(n-1)$
In the formula, $n$ is any term number and $a(n)$ is the $n^\text{th}$ term.
This formula allows us to simply plug in the number of the term we are interested in, and we will get the value of that term.
In order to find the fifth term, for example, we need to plug $n=5$ into the explicit formula.
\begin{aligned}a(\greenE 5)&=3+2(\greenE 5-1)\\\\ &=3+2\cdot4\\\\ &=3+8\\\\ &=11\end{aligned}
Cool! This is in fact the fifth term of $3, 5, 7,...$

1) Find $b(10)$ in the sequence given by $b(n) = -5+9(n-1)$.
$b(10)=$
\begin{aligned}b(\greenE{10})&=-5+9(\greenE{10}-1)\\\\ &=-5+9\cdot9\\\\ &=-5+81\\\\ &=76\end{aligned}

# Writing explicit formulas

Consider the arithmetic sequence $5,8,11,...$ The first term of the sequence is $\greenE5$ and the common difference is $\maroonC3$.
We can get any term in the sequence by taking the first term $\greenE5$ and adding the common difference $\maroonC3$ to it repeatedly. Check out, for example, the following calculations of the first few terms.
$n$Calculation for the $n^\text{th}$ term
$1$$\greenE{5}$$=\greenE{5}+0\cdot\maroonC{3}=5$
$2$$\greenE{5}\maroonC{+3}$$=\greenE{5}+1\cdot\maroonC{3}=8$
$3$$\greenE{5}\maroonC{+3+3}$$=\greenE{5}+2\cdot\maroonC{3}=11$
$4$$\greenE{5}\maroonC{+3+3+3}$$=\greenE{5}+3\cdot\maroonC{3}=14$
$5$$\greenE{5}\maroonC{+3+3+3+3}$$=\greenE{5}+4\cdot\maroonC{3}=17$
The table shows that we can get the $n^\text{th}$ term (where $n$ is any term number) by taking the first term $\greenE{5}$ and adding the common difference $\maroonC{3}$ repeatedly for $n\!-\!\!1$ times. This can be written algebraically as $\greenE{5}\maroonC{+3}(n-1)$.
In general, this is the standard explicit formula of an arithmetic sequence whose first term is $\greenE A$ and common difference is $\maroonC B$:
$\greenE A+\maroonC B(n-1)$

2) Write an explicit formula for the sequence $2, 9, 16,...$.
$d(n)=$
The general form is $\greenE A+\maroonC B(n-1)$ where $\greenE A$ is the first term and that $\maroonC B$ is the common difference.
• The first term is $\greenE 2$
• The common difference is $\maroonC 7$
Therefore, $d(n)=\greenE{2}\maroonC{+7}(n-1)$.
3) Write an explicit formula for the sequence $9, 5, 1,...$.
$e(n)=$
The general form is $\greenE A+\maroonC B(n-1)$ where $\greenE A$ is the first term and that $\maroonC B$ is the common difference.
• The first term is $\greenE 9$
• The common difference is $\maroonC{-4}$
Therefore, $e(n)=\greenE{9}\maroonC{-4}(n-1)$.
4) The explicit formula of a sequence is $f(n)=-6+2(n-1)$.
What is the first term of the sequence?
What is the common difference?
The general form is $\greenE A+\maroonC B(n-1)$ where $\greenE A$ is the first term and that $\maroonC B$ is the common difference.
Therefore, in the formula $\greenE{-6}\maroonC{+2}(n-1)$,
• the first term is $\greenE{-6}$, and
• the common difference is $\maroonC{2}$.

# Equivalent explicit formulas

Explicit formulas can come in many forms.
For example, the following are all explicit formulas for the sequence $3,5,7,...$
• $3+2(n-1)$ (this is the standard formula)
• $1+2n$
• $5+2(n-2)$
The formulas may look different, but the important thing is that we can plug an $n$-value and get the correct $n^\text{th}$ term (try for yourselves that the other formulas are correct!).
Different explicit formulas that describe the same sequence are called equivalent formulas.

# A common misconception

An arithmetic sequence may have different equivalent formulas, but it's important to remember that only the standard form gives us the first term and the common difference.
For example, the sequence $2, 8, 14,...$ has a first term of $\greenE 2$ and a common difference of $\maroonC 6$.
The explicit formula $\greenE 2\maroonC{+6}(n-1)$ describes this sequence, but the explicit formula $\greenE 2\maroonC{+6}n$ describes a different sequence.
$n\quad$$2+6(n-1)$$2+6n$
$1$$2$$8$
$2$$8$$14$
$3$$14$$20$
According to the table, we can see that the formula $2+6(n-1)$ describes the sequence $2,8,14,...$ while the formula $2+6n$ describes the sequence $8,14,20...$
In order to bring the formula $2+6(n-1)$ to an equivalent formula of the form $A+Bn$, we can expand the parentheses and simplify:
\begin{aligned}&\phantom{=}2+6(n-1)\\\\ &=2+6n-6\\\\ &=-4+6n\end{aligned}
Some people might prefer the formula $-4+6n$ over the equivalent formula $2+6(n-1)$, because it's shorter. The nice thing about the longer formula is that it gives us the first term.

5) Find all correct explicit formulas of the sequence $12, 7, 2,...$
• The first term is $\greenE{12}$
• The common difference is $\maroonC{-5}$
Therefore, a correct explicit formula is $\greenE{12}\maroonC{-5}(n-1)$.
This formula can be simplified as follows:
\begin{aligned}&\phantom{=}12-5(n-1)\\\\ &=12-5n+5\\\\ &=17-5n\end{aligned}
Therefore, this is a correct formula too.
In conclusion, these are the correct formulas:
• $17-5n$
• $12-5(n-1)$

### Challenge problems

6*) Find the $124^{\text{th}}$ term of the arithmetic sequence $199, 196, 193,...$
In order to answer, we will first find the explicit formula of the sequence.
Once we have the explicit formula, we can plug $n=124$ into that formula to find the answer.
• The first term is $\greenE{199}$
• The common difference is $\maroonC{-3}$
Therefore, a correct explicit formula is $g(n)=\greenE{199}\maroonC{-3}(n-1)$.
Now we plug $n=124$ into the formula.
\begin{aligned}g(124)&=199-3(124-1)\\\\ &=199-3\cdot 123\\\\ &=199-369\\\\ &=-170\end{aligned}
In conclusion, the $124^{\text{th}}$ term of the sequence is $-170$.
7*) The first term of an arithmetic sequence is $5$ and the tenth term is $59$.
What is the common difference?
Let's represent the common difference with $\maroonC B$.
• The first term is $\greenE 5$
• The common difference is $\maroonC B$
Therefore, an explicit formula for the sequence is $h(n)=\greenE 5+\maroonC B(n-1)$.
We are also given that $h(10)=59$. We can substitute this into the formula to obtain an equation where the only unknown is $\maroonC B$.
\begin{aligned}h(n)&=\greenE 5+\maroonC B(n-1)\\\\ 59&=\greenE 5+\maroonC B(10-1)&\gray{\text{Substitute }h(10)=59}\\\\ 54&=9\maroonC B\\\\ \maroonC 6&=\maroonC B \end{aligned}
Therefore, the explicit formula of the sequence is $\greenE 5\maroonC{+6}(n-1)$ and the common difference is $\maroonC 6$.