True or False A function is a relation between two sets $D$ and $R$ so that each element $x$ in the first set $D$ is related to exactly one element $y$ in the second set $R .

Check back soon!

A(n)_________is a function whose domain is the set of $A(n)$positive integers.

Check back soon!

True or False The notation $a_{5}$ represents the fifth term of a sequence.

Check back soon!

The sequence $a_{1}=5, a_{n}=3 a_{n-1}$ is an example of a _________sequence.

Check back soon!

The notation $a_{1}+a_{2}+a_{3}+\dots+a_{n}=\sum_{k=1}^{n} a_{k}$ is an example of notation.

Check back soon!

$$\text { True or False } \sum_{k=1}^{n} k=1+2+3+\cdots+n=\frac{n(n+1)}{2}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|s_{n}\right|=\{n\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|s_{n}\right|=\left\{n^{2}+1\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\quad\left\{a_{n}\right\}=\left\{\frac{n}{n+2}\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|b_{n}\right|=\left\{\frac{2 n+1}{2 n}\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|c_{n}\right|=\left\{(-1)^{n+1} n^{2}\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|d_{n}\right|=\left\{(-1)^{n-1}\left(\frac{n}{2 n-1}\right)\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|s_{n}\right|=\left\{\frac{2^{n}}{3^{n}+1}\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left\{s_{n}\right\}=\left\{\left(\frac{4}{3}\right)^{n}\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|t_{n}\right|=\left\{\frac{(-1)^{n}}{(n+1)(n+2)}\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|a_{n}\right|=\left\{\frac{3^{n}}{n}\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|b_{n}\right|=\left\{\frac{n}{e^{n}}\right\}$$

Check back soon!

Write down the first five terms of each sequence.

$$\left|c_{n}\right|=\left\{\frac{n^{2}}{2^{n}}\right\}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, $$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$\frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4}, \frac{1}{4 \cdot 5}, \dots$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \dots$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$1,-1,1,-1,1,-1, \ldots$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, 7, \frac{1}{8}, \dots$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$1,-2,3,-4,5,-6, \dots$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$2,-4,6,-8,10, \dots$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=2 ; \quad a_{n}=3+a_{n-1}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=3 ; \quad a_{n}=4-a_{n-1}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=-2 ; \quad a_{n}=n+a_{n-1}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=1 ; \quad a_{n}=n-a_{n-1}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=5 ; \quad a_{n}=2 a_{n-1}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=2 ; a_{n}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=3 ; \quad a_{n}=\frac{a_{n-1}}{n}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=-2 ; \quad a_{n}=n+3 a_{n-1}$$

Check back soon!

$$a_{1}=-2 ; \quad a_{n}=n+3 a_{n-1}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=1 ; \quad a_{2}=2 ; \quad a_{n}=a_{n-1} \cdot a_{n-2}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=-1 ; \quad a_{2}=1 ; \quad a_{n}=a_{n-2}+n a_{n-1}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=A ; \quad a_{n}=a_{n-1}+d$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=A ; \quad a_{n}=r a_{n-1}, \quad r \neq 0$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$y=\sqrt{2} ; \quad a_{n}=\sqrt{2+a_{n-1}}$$

Check back soon!

A sequence is defined recursively. Write down the first five terms.

$$a_{1}=\sqrt{2} ; \quad a_{n}=\sqrt{\frac{a_{n-1}}{2}}$$

Check back soon!

Express each sum using summation notation.

$$1^{3}+2^{3}+3^{3}+\dots+8^{3}$$

Check back soon!

Express each sum using summation notation.

$$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+\frac{13}{13+1}$$

Check back soon!

Express each sum using summation notation.

$$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots+(-1)^{6}\left(\frac{1}{3^{6}}\right)$$

Check back soon!

Express each sum using summation notation.

$$\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\dots+(-1)^{12}\left(\frac{2}{3}\right)^{11}$$

Check back soon!

Express each sum using summation notation.

$$3+\frac{3^{2}}{2}+\frac{3^{3}}{3}+\dots+\frac{3^{n}}{n}$$

Check back soon!

Express each sum using summation notation.

$$\frac{1}{e}+\frac{2}{e^{2}}+\frac{3}{e^{3}}+\dots+\frac{n}{e^{n}}$$

Check back soon!

Express each sum using summation notation.

$$a+(a+d)+(a+2 d)+\cdots+(a+n d)$$

Check back soon!

Express each sum using summation notation.

$$a+a r+a r^{2}+\dots+a r^{n-1}$$

Check back soon!

John has a balance of $\$ 3000$ on his Discover card that charges $1 \%$ interest per month on any unpaid balance. John can afford to pay $\$ 100$ toward the balance each month. His balance each month after making a $\$ 100$ payment is given by the recursively defined sequence.

$$B_{0}=\$ 3000 \quad B_{n}=1.01 B_{n-1}-100$$

Determine John's balance after making the first payment. That is, determine $B_{1}$.

Check back soon!

Trout Population A pond currently has 2000 trout in it.A fish hatchery decides to add an additional 20 trout each month. In addition, it is known that the trout population is growing $3 \%$ per month. The size of the population after $n$ months is given by the recursively defined sequence

$$p_{0}=2000 \quad p_{n}=1.03 p_{n-1}+20$$

How many trout are in the pond after two months? That is, what is $p_{2} ?$.

Check back soon!

Car Loans Phil bought a car by taking out a loan for $\$ 18,500$ at $0.5 \%$ interest per month. Phil's normal monthly payment is $\$ 434.47$ per month, but he decides that he can afford to pay $\$ 100$ extra toward the balance each month. His balance each month is given by the recursively defined sequence

$$B_{0}=\$ 18,500 \quad B_{m}=1.005 B_{n-1}-534.47$$

Determine Phil's balance after making the first payment. That is, determine $B_{1}$.

Check back soon!

The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of Sollutant as a result of industrial waste and that $10 \%$ of the Sollutant present is neutralized by solar oxidation every Jyear. The EPA imposes new pollution control laws that eresult in 15 tons of new pollutant entering the lake each Jyear. The amount of pollutant in the lake after $n$ years is esiven by the recursively defined sequence.

$$p_{0}=250 \quad p_{n}=0.9 p_{n-1}+15$$

Determine the amount of pollutant in the lake after 2 years. That is, determine $p_{2}$.

Check back soon!

A colony of rabbits begins with one pair of mature rabbits, which will produce a pair of offspring (one male, one female) each month. Assume that all rabbits mature in 1 month and produce a pair of offspring (one male, one female) after 2 months. If no rabbits ever die, how many pairs of mature rabbits are there after 7 months? [Hint: A Fibonacci sequence models this colony. Do you see why?

(graph can't copy)

Check back soon!

Let.

$$u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$

define the $n$ th term of a sequence.

(a) Show that $u_{1}=1$ and $u_{2}=1$

(b) Show that $u_{n+2}=u_{n+1}+u_{n}$

(c) Draw the conclusion that $\left\{u_{n}\right\}$ is a Fibonacci sequence.

Check back soon!

The triangular array shown (called Pascal's triangle) using diagonal lines as indicated. Find the sum of the numbers in each diagonal row. Do you recognize this sequence?

(graph can't copy)

Check back soon!

Fibonacci Sequence Use the result of Problem 86 to do the following problems:

(a) Write the first 11 terms of the Fibonacci sequence.

(b) Write the first 10 terms of the ratio $\frac{u_{n+1}}{u_{n}}$

(c) As $n$ gets large, what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the façade of the Parthenon was constructed using the golden ratio.

(d) Write down the first 10 terms of the ratio $\frac{u_{n}}{u_{n+1}}$

(e) As $n$ gets large, what number does the ratio approach? This number is referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid.

Check back soon!

Approximating $f(x)=e^{x} \quad$ In calculus, it can be shown that $$f(x)=e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}$$We can approximate the value of $f(x)=e^{x}$ for any $x$ using the following sum

$$f(x)=e^{x} \approx \sum_{k=0}^{n} \frac{x^{k}}{k !}$$.

for some $n .$

(a) Approximate $f(1.3)$ with $n=4$

(b) Approximate $f(1.3)$ with $n=7$

(c) Use a calculator to approximate $f(1.3)$

"(d) Using trial and error along with a graphing utility's SEQuence mode, determine the value of $n$ required to approximate $f(1.3)$ correct to eight decimal places

Check back soon!

Approximating $f(x)=e^{x} \quad$ Refer to Problem 89

(a) Approximate $f(-2.4)$ with $n=3$

(b) Approximate $f(-2.4)$ with $n=6$

- (c) Use a calculator to approximate $f(-2.4)$

a d) Using trial and error along with a graphing utility's SEOuence mode, determine the value of $n$ required to approximate $f(-2.4)$ correct to eight decimal places.

Check back soon!

Bode's Law In $1772,$ Johann Bode published the following formula for predicting the mean distances, in astronomical units (AU), of the planets from the sun:

$$a_{1}=0.4 \quad a_{n}=0.4+0.3 \cdot 2^{n-2}, n \geq 2$$ where $n$ is the number of the planet from the sun.

(a) Determine the first eight terms of this sequence.

(b) At the time of Bode's publication, the known planets were Mercury (0.39 AU), Venus (0.72 AU), Earth (1 AU), Mars (1.52 AU), Jupiter (5.20 AU), and Saturn (9.54 AU). How do the actual distances compare to the terms of the sequence?

(c) The planet Uranus was discovered in 1781 and the asteroid Ceres was discovered in $1801 .$ The mean orbital distances from the sun to Uranus and Ceres" are $19.2 \mathrm{AU}$ and $2.77 \mathrm{AU}$, respectively. How well do these values fit within the sequence?

(d) Determine the ninth and tenth terms of Bode's sequence.

(e) The planets Neptune and Pluto" were discovered in 1846 and $1930,$ respectively. Their mean orbital distances from the sun are $30.07 \mathrm{AU}$ and $39.44 \mathrm{AU}$, respectively. How do these actual distances compare to the terms of the sequence?

(f) On July $29,2005,$ NASA announced the discovery of a dwarf planet" $(n=11),$ which has been named Eris Use Bode's Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun.

Check back soon!

Show that.

$$1+2+\cdots+(n-1)+n=\frac{n(n+1)}{2}$$

let $$S=1+2+\cdots+(n-1)+n$$

$$S=n+(n-1)+(n-2)+\cdots+1$$

Add these equations Then $$2 S=[1+n]+[2+(n-1)]+\cdots+[n+1]$$

n terms in brsckets Now complete the derivation.

Check back soon!

A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence

$$a_{0}=k \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)$$

Where $k$ is an imulal guess as to the value of the square root. Use $t$ recursive formula to approximate the following square roots by finding $a_{5} .$ Compare this result to the value provided by your calculator.

$$\sqrt{5}$$

Check back soon!

A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence

$$a_{0}=k \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)$$

Where $k$ is an imulal guess as to the value of the square root. Use $t$ recursive formula to approximate the following square roots by finding $a_{5} .$ Compare this result to the value provided by your calculator.

$$\sqrt{8}$$

Check back soon!

A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence

$$a_{0}=k \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)$$

Where $k$ is an imulal guess as to the value of the square root. Use $t$ recursive formula to approximate the following square roots by finding $a_{5} .$ Compare this result to the value provided by your calculator.

$$\sqrt{21}$$

Check back soon!

A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence

$$a_{0}=k \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)$$

Where $k$ is an imulal guess as to the value of the square root. Use $t$ recursive formula to approximate the following square roots by finding $a_{5} .$ Compare this result to the value provided by your calculator.

$$\sqrt{89}$$

Check back soon!

Triangular Numbers $A$ triangular number is a term of the sequence

$$u_{1}=1, \quad u_{n+1}=u_{n}+(n+1)$$

Write down the first seven triangular numbers.

Check back soon!

For the sequence given in Problem $97,$ show that

$$u_{n+1}=\frac{(n+1)(n+2)}{2}$$.

Check back soon!

For the sequence given in Problem $97,$ show that

$$u_{n+1}+u_{n}=(n+1)^{2}$$

Check back soon!

Investigate various applications that lead to a Fibonacci sequence, such as art, architecture, or financial markets. Write an essay on these applications.

Check back soon!

Write a paragraph that explains why the numbers found in Problem 97 are called triangular.

Check back soon!