##### 1.

\(a_n = \dfrac{n}{n^2+1}\)

- A function whose inputs are a set of successive positive integers is called a sequence.
- The output values are called the terms of the sequence.
- A formula in terms of \(n\) for the \(n^{\text{th}}\) term of a sequence is called the general term of the sequence.
- A sequence is defined recursively if each term of the sequence is defined in terms of its predecessors.
- A sequence in which each term can be obtained from the previous term by adding a fixed amount is called an arithmetic sequence.
- The fixed amount between two successive terms of an arithmetic sequence is called the common difference.
- An arithmetic sequence defines a linear function of \(n\text{.}\)
- The \(n^{\text{th}}\) term of an arithmetic sequence is \(~a_n = a + (n-1)d\text{.}\)
- A sequence in which each term can be obtained from the previous term by multiplying by a fixed amount is called a geometric sequence.
- The ratio of two successive terms of a geometric sequence is called the common ratio.
- A geometric sequence defines an exponential function of \(n\text{.}\)
- The \(n^{\text{th}}\) term of a geometric sequence is \(~a_n = ar^{n-1}\text{.}\)
- The sum of the terms of a sequence is called a series.
- The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is\begin{equation*} S_n = \dfrac{n}{2} (a_1 + a_n) \end{equation*}
- The sum \(S_n\) of the first \(n\) terms of a geometric sequence is\begin{equation*} S_n = \dfrac{a_{n+1} - a_1}{r-1} \end{equation*}
- We can use sigma notation to denote a series.
- It is possible to add infinitely many terms and arrive at a finite sum if the terms are small enough.
- The sum of an infinite geometric series \(~~\displaystyle{\sum_{k=0}^{\infty} ar^{k-1}}~~\) is\begin{equation*} S_{\infty} = \dfrac{a}{1-r}~~~~~~\text{if}~~~~~~-1 \lt r \lt 1 \end{equation*}
- If \(\abs{r} \ge 1\) in an infinite geometric series, the series does not have a sum.
- The binomial coefficient \(~_nC_k~\) is the coefficient of the term containing \(b^k\) in the expansion of \((a+b)^n\text{.}\)
- The numbers in Pascal’s triangle are the binomial coefficients. Specifically, the number in the \(k^{\text{th}}\) position (starting with \(k=0\)) of the \(n^{\text{th}}\) row of the triangle is \(~_nC_k~\text{.}\)
- If \(n\) is a positive integer,\begin{equation*} n! = n \cdot (n-1) \cdot (n-2) \cdot \cdots \cdot 3\cdot 2 \cdot 1 \end{equation*}
- For \(n \ge 0\) and \(0 \le k \le n\text{,}\)\begin{equation*} ~_nC_k~ = \dfrac{n!}{(n-k)!~k!} \end{equation*}
- The Binomial Theorem\begin{equation*} (a+b)^n = ~~\displaystyle{\sum_{k=0}^{n} ~_nC_k~ a^{n-k}b^k} \end{equation*}

For Problems 1–2, find the first four terms in the sequence whose general term is given.

\(a_n = \dfrac{n}{n^2+1}\)

\(b_n = \dfrac{(-1)^{n-1}}{n}\)

For Problems 3–4, find the first five terms in the recursively defined sequence.

\(c_1 = 5;~~c_{n+1}=c_n-3\)

\(d_1=1;~~d_{n+1} = \dfrac{-3}{4}d_n\)

For Problems 5–8,

- Find the first four terms of the sequence.
- Determine equations to define the sequence recursively.

Rick purchased a sailboat for $1800. How much is it worth after \(n\) years if it depreciates in value by 12% each year?

Sally earns $24,000 per year. If she receives a 6% raise each year, what will her salary be after \(n\) years?

To fight off an infection, Garrison receives a 30-milliliter dose of an antibiotic followed by doses of 15-milliliters at regular intervals. Between doses, Garrison’s kidneys body removes 25% of the antibiotic that was present after the previous dose. How much of the antibiotic is present after \(n\) doses?

Opal joined the Weight Losers Club. At the end of the first meeting she weighed 187 pounds, and she lost 2 pounds from one meeting to the next. How much does Opal weigh after \(n\) meetings?

For Problems 9–18, find the indicated term for the sequence.

\(x_n = (-1)^n (n-2)^2;~~x_7\)

\(y_n = \sqrt{n^3 - 2};~~y_3\)

The tenth term in the arithmetic sequence that begins \(~{-4}, 0, \cdots\text{.}\)

The sixth term in the arithmetic sequence that begins \(~x-a,~ x+a, \cdots\text{.}\)

The eighth term in the geometric sequence that begins \(~\dfrac{16}{27}, \dfrac{-8}{9}, \dfrac{2}{3}, \cdots\)

The fifth term in the geometric sequence with third term \(\dfrac{-2}{3}\) and sixth term \(\dfrac{16}{81}.\)

The twenty-third term of the arithmetic sequence \(~{-84}, -74, -64, \cdots\)

The ninth term of the arithmetic sequence \(~\dfrac{-1}{2}, 1, \dfrac{5}{2}, \cdots\)

The first term of an arithmetic sequence is 8 and the twenty-eighth term is 89. Find the twenty-first term.

What term in the arithmetic sequence \(~5, 2, -1, \cdots~\) is \(~{-37}\text{?}\)

For Problems 19–28, identify each sequence as arithmetic, geometric, or neither. If the sequence is arithmetic or geometric, find the next four terms of the sequence and a non-recursive expression for the general term.

\(-1, \dfrac{1}{2}, \dfrac{-1}{4}, \dfrac{1}{8}, \cdots\)

\(12, 9, -3, 1, \cdots\)

\(6, 1, -4, 9, \cdots\)

\(1, -4, 16, -64, \cdots\)

\(-1, 2, -4, 8, \cdots\)

\(\dfrac{2}{3}, \dfrac{1}{2}, \dfrac{3}{8}, \dfrac{9}{32}, \cdots\)

First term \(3, \) common difference \(-4\)

First term \(\dfrac{1}{4}\text{,}\) common difference \(\dfrac{1}{2}\)

First term \(12, \) common ratio \(-4\)

First term \(6, \) common ratio \(\dfrac{1}{3}\)

For Problems 29–30, write the sum in expanded form.

\(~~\displaystyle{\sum_{k=2}^{5} ~k(k-1)}\)

\(~~\displaystyle{\sum_{j=2}^{\infty} ~\dfrac{j}{2j-1}}\)

For Problems 31–32, write the sum using sigma notation.

The first 12 terms of \(~1, 3, 7, \cdots 2^k-1, \cdots\)

The fourth through fifteenth terms of \(~x, 4x^2, 9x^3, \cdots, k^2x^k, \cdots\)

For Problems 33–42, identify the series as arithmetic, geometric, or neither, then evaluate.

The sum of the first 12 terms of the sequence \(~a_n = 3n-2\)

The sum of the first 20 terms of the sequence \(~b_n = 1.4 + 0.1n\)

\(~~\displaystyle{\sum_{i=1}^{6} ~(3i-1)}\)

\(~~\displaystyle{\sum_{k=1}^{12} ~\left(\dfrac{2}{3}k-1\right)}\)

\(~~\displaystyle{\sum_{j=1}^{5} ~\left(\dfrac{1}{3}\right)^j}\)

\(~~\displaystyle{\sum_{k=1}^{6} ~2^{k-1}}\)

\(~~\displaystyle{\sum_{n=1}^{5} ~(-1)^{n}(n+1)}\)

\(~~\displaystyle{\sum_{n=1}^{4} ~\dfrac{n}{n+1}}\)

\(~-3 + 2 + \left(\dfrac{-4}{3}\right) + \left(\dfrac{8}{9}\right) + \cdots \)

\(~~\displaystyle{\sum_{1=1}^{\infty} ~3(\dfrac{1}{3})^{i-1}}\)

A rubber ball is dropped from a height of 12 feet and returns two-thirds of its previous height on each bounce. How high does the ball bounce after hitting the floor for the fourth time?

The property taxes on the Hardesty’s family home were $840 in 2014. If the taxes increase by 2% each year, what will the taxes be in 2020?

- Find the sum of all integral multiples of 6 between 10 and 100.
- Write the sum in (a) using sigma notation.

Kathy planted a 7-foot silver maple tree in 2015. If the tree grows 1.3 feet each year, in what year will it be 20 feet tall?

A rubber ball is dropped from a height of 12 feet and returns three-fourths of its previous height on each bounce. Approximately what is the total distance the ball travels before coming to rest?

Suppose you will be paid 1¢ on the first day of June, 2¢ on the second, 4¢ on the third, etc., so that each new day you are paid twice what your received the previous day. What would be the total amount you will receive in the month of June?

For Problems 49–50, find a common fraction equivalent to the repeating decimal.

\(3.222\overline{2}\)

\(0.41818\overline{18}\)

For Problems 51–52, write the power in expanded form.

\((x-2)^5\)

\(\left(\dfrac{x}{2}-y\right)^4\)

For Problems 53–58, evaluate.

\(\dfrac{6!}{3!(6-3)!}\)

\(\dfrac{9!}{5!(9-5)!}\)

\(~_7C_2~\)

\(~_{16}C_{14}~\)

\(~~\displaystyle{\sum_{k=0}^{5} ~_5C_k~}\)

\(~~\displaystyle{\sum_{k=0}^{6} ~_6C_k~ 1.4^{6-k}0.6^k}\)

For Problems 59–60, find the coefficient of the indicated term.

\((x-2y)^9;~~x^6y^3\)

\(\left(\dfrac{x}{2} - 3\right)^8;~~x\)