College Algebra

Michael Sullivan

Chapter 9

Sequences; induction; the Binomial Theorem

Educators


Problem 1

For the function $f(x)=\frac{x-1}{x},$ find $f(2)$ and $f(3)$.

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Problem 2

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 .

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Problem 3

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

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Problem 4

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

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Problem 5

If $n \geq 0$ is an integer, then $n !=$ _________when $n=2$.

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Problem 6

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

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Problem 7

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

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Problem 8

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

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Problem 9

Evaluate each factorial expression.
$$9.10 !$$

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Problem 10

Evaluate each factorial expression.
$$10.9 !$$

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Problem 11

Evaluate each factorial expression.
$$\frac{91}{6 !}$$

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Problem 12

Evaluate each factorial expression.
$$\frac{12 !}{10 !}$$

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Problem 13

Evaluate each factorial expression.
$$\frac{3 ! 7 !}{4 !}$$

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Problem 14

Evaluate each factorial expression.
$$\frac{5 ! 8 !}{3 !}$$

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Problem 15

Write down the first five terms of each sequence.
$$\left|s_{n}\right|=\{n\}$$

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Problem 16

Write down the first five terms of each sequence.
$$\left|s_{n}\right|=\left\{n^{2}+1\right\}$$

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Problem 17

Write down the first five terms of each sequence.
$$\quad\left\{a_{n}\right\}=\left\{\frac{n}{n+2}\right\}$$

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Problem 18

Write down the first five terms of each sequence.
$$\left|b_{n}\right|=\left\{\frac{2 n+1}{2 n}\right\}$$

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Problem 19

Write down the first five terms of each sequence.
$$\left|c_{n}\right|=\left\{(-1)^{n+1} n^{2}\right\}$$

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Problem 20

Write down the first five terms of each sequence.
$$\left|d_{n}\right|=\left\{(-1)^{n-1}\left(\frac{n}{2 n-1}\right)\right\}$$

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Problem 21

Write down the first five terms of each sequence.
$$\left|s_{n}\right|=\left\{\frac{2^{n}}{3^{n}+1}\right\}$$

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Problem 22

Write down the first five terms of each sequence.
$$\left\{s_{n}\right\}=\left\{\left(\frac{4}{3}\right)^{n}\right\}$$

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Problem 23

Write down the first five terms of each sequence.
$$\left|t_{n}\right|=\left\{\frac{(-1)^{n}}{(n+1)(n+2)}\right\}$$

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Problem 24

Write down the first five terms of each sequence.
$$\left|a_{n}\right|=\left\{\frac{3^{n}}{n}\right\}$$

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Problem 25

Write down the first five terms of each sequence.
$$\left|b_{n}\right|=\left\{\frac{n}{e^{n}}\right\}$$

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Problem 26

Write down the first five terms of each sequence.
$$\left|c_{n}\right|=\left\{\frac{n^{2}}{2^{n}}\right\}$$

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Problem 27

A sequence is defined recursively. Write down the first five terms.
$$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, $$

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Problem 28

A sequence is defined recursively. Write down the first five terms.
$$\frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4}, \frac{1}{4 \cdot 5}, \dots$$

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Problem 29

A sequence is defined recursively. Write down the first five terms.
$$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$$

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Problem 30

A sequence is defined recursively. Write down the first five terms.
$$\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \dots$$

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Problem 31

A sequence is defined recursively. Write down the first five terms.
$$1,-1,1,-1,1,-1, \ldots$$

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Problem 32

A sequence is defined recursively. Write down the first five terms.
$$1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, 7, \frac{1}{8}, \dots$$

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Problem 33

A sequence is defined recursively. Write down the first five terms.
$$1,-2,3,-4,5,-6, \dots$$

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Problem 34

A sequence is defined recursively. Write down the first five terms.
$$2,-4,6,-8,10, \dots$$

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Problem 35

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=2 ; \quad a_{n}=3+a_{n-1}$$

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Problem 36

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=3 ; \quad a_{n}=4-a_{n-1}$$

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Problem 37

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=-2 ; \quad a_{n}=n+a_{n-1}$$

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Problem 38

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=1 ; \quad a_{n}=n-a_{n-1}$$

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Problem 39

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=5 ; \quad a_{n}=2 a_{n-1}$$

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Problem 40

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=2 ; a_{n}$$

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Problem 41

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=3 ; \quad a_{n}=\frac{a_{n-1}}{n}$$

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Problem 42

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=-2 ; \quad a_{n}=n+3 a_{n-1}$$

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Problem 42

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=-2 ; \quad a_{n}=n+3 a_{n-1}$$

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Problem 43

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=1 ; \quad a_{2}=2 ; \quad a_{n}=a_{n-1} \cdot a_{n-2}$$

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Problem 44

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=-1 ; \quad a_{2}=1 ; \quad a_{n}=a_{n-2}+n a_{n-1}$$

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Problem 45

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=A ; \quad a_{n}=a_{n-1}+d$$

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Problem 46

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=A ; \quad a_{n}=r a_{n-1}, \quad r \neq 0$$

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Problem 47

A sequence is defined recursively. Write down the first five terms.
$$y=\sqrt{2} ; \quad a_{n}=\sqrt{2+a_{n-1}}$$

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Problem 48

A sequence is defined recursively. Write down the first five terms.
$$a_{1}=\sqrt{2} ; \quad a_{n}=\sqrt{\frac{a_{n-1}}{2}}$$

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Problem 49

Write out each sum.
$$\sum_{k=1}^{n}(k+2)$$

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Problem 50

Write out each sum.
$$\sum_{i=1}^{n}(2 k+1)$$

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Problem 51

Write out each sum.
$$\sum_{k=1}^{n} \frac{k^{2}}{2}$$

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Problem 52

Write out each sum.
$$\sum_{k=1}^{n}(k+1)^{2}$$

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Problem 53

Write out each sum.
$$\sum_{k=0}^{n} \frac{1}{3^{k}}$$

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Problem 54

Write out each sum.
$$\hat{\Sigma}\left(\frac{3}{2}\right)^{k}$$

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Problem 55

Write out each sum.
$$\sum_{k=1}^{n-1} \frac{1}{3^{k+1}}$$

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Problem 56

Write out each sum.
$$\sum_{k=0}^{n-1}(2 k+1)$$

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Problem 57

Write out each sum.
$$\sum_{k=2}^{n}(-1)^{k} \ln k$$

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Problem 58

Write out each sum.
$$\sum_{k=1}^{n}(-1)^{k+1} 2^{k}$$

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Problem 59

Express each sum using summation notation.
$$1+2+3+\dots+20$$

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Problem 60

Express each sum using summation notation.
$$1^{3}+2^{3}+3^{3}+\dots+8^{3}$$

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Problem 61

Express each sum using summation notation.
$$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+\frac{13}{13+1}$$

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Problem 62

Express each sum using summation notation.
$$1+3+5+7+\dots+[2(12)-1]$$

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Problem 63

Express each sum using summation notation.
$$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots+(-1)^{6}\left(\frac{1}{3^{6}}\right)$$

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Problem 64

Express each sum using summation notation.
$$\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\dots+(-1)^{12}\left(\frac{2}{3}\right)^{11}$$

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Problem 65

Express each sum using summation notation.
$$3+\frac{3^{2}}{2}+\frac{3^{3}}{3}+\dots+\frac{3^{n}}{n}$$

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Problem 66

Express each sum using summation notation.
$$\frac{1}{e}+\frac{2}{e^{2}}+\frac{3}{e^{3}}+\dots+\frac{n}{e^{n}}$$

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Problem 67

Express each sum using summation notation.
$$a+(a+d)+(a+2 d)+\cdots+(a+n d)$$

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Problem 68

Express each sum using summation notation.
$$a+a r+a r^{2}+\dots+a r^{n-1}$$

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Problem 69

Find the sum of each sequence.
$$\sum_{k=1}^{\infty} 5$$

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Problem 70

Find the sum of each sequence.
$$\sum_{k=1}^{50} 8$$

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Problem 71

Find the sum of each sequence.
$$\sum_{k=1}^{\infty} k$$

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Problem 72

Find the sum of each sequence.
$$\sum_{k=1}^{24}(-k)$$

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Problem 73

Find the sum of each sequence.
$$\sum_{k=1}^{24}(-k)$$

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Problem 74

Find the sum of each sequence.
$$\sum_{k=1}^{26}(3 k-7)$$

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Problem 75

Find the sum of each sequence.
$$\sum_{k=1}^{16}\left(k^{2}+4\right)$$

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Problem 76

Find the sum of each sequence.
$$\sum_{k=0}^{14}\left(k^{2}-4\right)$$

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Problem 77

Find the sum of each sequence.
$$\sum_{t=1}^{\infty}(2 k)$$

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Problem 78

Find the sum of each sequence.
$$\sum_{i}(-3 k)$$

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Problem 79

Find the sum of each sequence.
$$\sum_{i=1}^{\infty} k^{3}$$

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Problem 80

Find the sum of each sequence.
$$\sum_{i=1}^{2 i} k^{3}$$

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Problem 81

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 $B_{1}$.

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Problem 82

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 $p_{2} ?$.

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Problem 83

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 $B_{1}$.

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Problem 84

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 $p_{2}$.

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Problem 85

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)

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Problem 86

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.

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Problem 87

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)

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Problem 88

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.

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Problem 89

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

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Problem 90

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.

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Problem 91

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.

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Problem 92

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.

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Problem 93

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.
$$\sqrt{5}$$

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Problem 94

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.
$$\sqrt{8}$$

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Problem 95

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.
$$\sqrt{21}$$

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Problem 96

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.
$$\sqrt{89}$$

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Problem 97

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.

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Problem 98

For the sequence given in Problem $97,$ show that
$$u_{n+1}=\frac{(n+1)(n+2)}{2}$$.

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Problem 99

For the sequence given in Problem $97,$ show that
$$u_{n+1}+u_{n}=(n+1)^{2}$$

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Problem 100

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

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Problem 101

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

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