You buy new furniture at zero percent interest on a monthly installment plan. The total of your furniture is $4800. The following sequence shows the balance you still owe on the furniture at the beginning of each month. How would you write a general rule for the sequence?
4800, 4600, 4400, 4200,...
Guidance
In the previous concept we wrote a recursive rule to find the next term in a sequence. Recursive rules can help us generate multiple sequential terms in a sequence but are not helpful in determining a particular single term. Consider the sequence:
Example A
Write the first three terms, the
Solution: We can find each of these terms by replacing
Calculator: These terms can also be found using a graphing calculator. First press
Example B
Write a general rule for the sequence:
Solution: The previous example illustrates how a general rule maps a term number directly to the term value. Another way to say this is that the general rule expresses the
Looking at the terms and term numbers together helps us to see that each term is the result of multiplying the term number by 5. The general rule is

1  2  3  4 


5  10  15  20 
Example C
Find the
Solution: Let’s make the table again to begin to analyze the relationship between the term number and the term value.

1  2  3  4 


0  2  6  12 





This time the pattern is not so obvious. To start, write each term as a product of the term number and a second factor. Then it can be observed that the second factor is always one less that the term number and the general rule can be written as
Intro Problem Revisit Let’s put the terms in the sequence in a table with their term numbers to help identify the rule.

1  2  3  4 


4800  4600  4400  4200 
Looking at the terms and term numbers together helps us to see that each term is the result of subtracting 200 times one less than the term from the first term. The general rule is
Guided Practice
1. Given the general rule:
2. Write the general rule for the sequence:
3. Write the general rule and find the
Answers
1. Plug in the term numbers as shown:
2. Put the values in a table with the term numbers and see if there is a way to write the term as a function of the term number.

1  2  3  4 


4  5  6  7 





Each term appears to be the result of adding three to the term number. Thus, the general rule is \begin{align*}a_n=n+3\end{align*}
3. Put the values in a table with the term numbers and see if there is a way to write the term as a function of the term number.
\begin{align*}n\end{align*}  1  2  3  4  5 

\begin{align*}a_n\end{align*}  1  10  3  8  15 
\begin{align*}n(?)\end{align*}  \begin{align*}(1)(1)\end{align*}  \begin{align*}(2)(0)\end{align*}  \begin{align*}(3)(1)\end{align*}  \begin{align*}(4)(2)\end{align*}  \begin{align*}(5)(3)\end{align*} 
Each term appears to be the result of multiplying the term number by two less than the term number. Thus, the general rule is \begin{align*}a_n=n(n2)\end{align*}.
Vocabulary
 \begin{align*}n^{th}\end{align*} term or general rule
 A formula which relates the term to the term number and thus can be used to calculate any term in a sequence whether or not any terms are known.
Explore More
Use the \begin{align*}n^{th}\end{align*} term rule to generate the indicated terms in each sequence.
 \begin{align*}2n+7\end{align*}, terms \begin{align*}15\end{align*} and the \begin{align*}10^{th}\end{align*} term.
 \begin{align*}5n1\end{align*}, terms \begin{align*}13\end{align*} and the \begin{align*}50^{th}\end{align*} term.
 \begin{align*}2^n1\end{align*}, terms \begin{align*}13\end{align*} and the \begin{align*}10^{th}\end{align*} term.
 \begin{align*}\left(\frac{1}{2}\right)^n\end{align*}, terms \begin{align*}13\end{align*} and the \begin{align*}8^{th}\end{align*} term.
 \begin{align*}\frac{n(n+1)}{2}\end{align*}, terms \begin{align*}14\end{align*} and the \begin{align*}20^{th}\end{align*} term.
Use your calculator to generate the first 5 terms in each sequence. Use MATH > FRAC, on your calculator to convert decimals to fractions.
 \begin{align*}4n3\end{align*}
 \begin{align*} \frac{1}{2}n+5\end{align*}
 \begin{align*}\left(\frac{2}{3}\right)^n+1\end{align*}
 \begin{align*}2n(n1)\end{align*}
 \begin{align*}\frac{n(n+1)(2n+1)}{6}\end{align*}
Write the \begin{align*}n^{th}\end{align*} term rule for the following sequences.
 \begin{align*}3,5,7,9,\ldots\end{align*}
 \begin{align*}1,7,25,79,\ldots\end{align*}
 \begin{align*}6,14,24,36,\ldots\end{align*}
 \begin{align*}6,5,4,3,\ldots\end{align*}
 \begin{align*}2,5,9,14,\ldots\end{align*}