College Algebra

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Problem 1

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

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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 $R . Check back soon! Problem 3 A(n)_________is a function whose domain is the set of$A(n)$positive integers. Check back soon! Problem 4 True or False The notation$a_{5}$represents the fifth term of a sequence. Check back soon! Problem 5 If$n \geq 0$is an integer, then$n !=$_________when$n=2$. Check back soon! Problem 6 The sequence$a_{1}=5, a_{n}=3 a_{n-1}$is an example of a _________sequence. Check back soon! Problem 7 The notation$a_{1}+a_{2}+a_{3}+\dots+a_{n}=\sum_{k=1}^{n} a_{k}$is an example of notation. Check back soon! Problem 8 $$\text { True or False } \sum_{k=1}^{n} k=1+2+3+\cdots+n=\frac{n(n+1)}{2}$$ Check back soon! Problem 9 Evaluate each factorial expression. $$9.10 !$$ Check back soon! Problem 10 Evaluate each factorial expression. $$10.9 !$$ Check back soon! Problem 11 Evaluate each factorial expression. $$\frac{91}{6 !}$$ Check back soon! Problem 12 Evaluate each factorial expression. $$\frac{12 !}{10 !}$$ Check back soon! Problem 13 Evaluate each factorial expression. $$\frac{3 ! 7 !}{4 !}$$ Check back soon! Problem 14 Evaluate each factorial expression. $$\frac{5 ! 8 !}{3 !}$$ Check back soon! Problem 15 Write down the first five terms of each sequence. $$\left|s_{n}\right|=\{n\}$$ Check back soon! Problem 16 Write down the first five terms of each sequence. $$\left|s_{n}\right|=\left\{n^{2}+1\right\}$$ Check back soon! Problem 17 Write down the first five terms of each sequence. $$\quad\left\{a_{n}\right\}=\left\{\frac{n}{n+2}\right\}$$ Check back soon! Problem 18 Write down the first five terms of each sequence. $$\left|b_{n}\right|=\left\{\frac{2 n+1}{2 n}\right\}$$ Check back soon! Problem 19 Write down the first five terms of each sequence. $$\left|c_{n}\right|=\left\{(-1)^{n+1} n^{2}\right\}$$ Check back soon! Problem 20 Write down the first five terms of each sequence. $$\left|d_{n}\right|=\left\{(-1)^{n-1}\left(\frac{n}{2 n-1}\right)\right\}$$ Check back soon! Problem 21 Write down the first five terms of each sequence. $$\left|s_{n}\right|=\left\{\frac{2^{n}}{3^{n}+1}\right\}$$ Check back soon! Problem 22 Write down the first five terms of each sequence. $$\left\{s_{n}\right\}=\left\{\left(\frac{4}{3}\right)^{n}\right\}$$ Check back soon! Problem 23 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! Problem 24 Write down the first five terms of each sequence. $$\left|a_{n}\right|=\left\{\frac{3^{n}}{n}\right\}$$ Check back soon! Problem 25 Write down the first five terms of each sequence. $$\left|b_{n}\right|=\left\{\frac{n}{e^{n}}\right\}$$ Check back soon! Problem 26 Write down the first five terms of each sequence. $$\left|c_{n}\right|=\left\{\frac{n^{2}}{2^{n}}\right\}$$ Check back soon! Problem 27 A sequence is defined recursively. Write down the first five terms. $$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5},$$ Check back soon! Problem 28 A sequence is defined recursively. Write down the first five terms. $$\frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4}, \frac{1}{4 \cdot 5}, \dots$$ Check back soon! Problem 29 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! Problem 30 A sequence is defined recursively. Write down the first five terms. $$\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \dots$$ Check back soon! Problem 31 A sequence is defined recursively. Write down the first five terms. $$1,-1,1,-1,1,-1, \ldots$$ Check back soon! Problem 32 A sequence is defined recursively. Write down the first five terms. $$1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, 7, \frac{1}{8}, \dots$$ Check back soon! Problem 33 A sequence is defined recursively. Write down the first five terms. $$1,-2,3,-4,5,-6, \dots$$ Check back soon! Problem 34 A sequence is defined recursively. Write down the first five terms. $$2,-4,6,-8,10, \dots$$ Check back soon! Problem 35 A sequence is defined recursively. Write down the first five terms. $$a_{1}=2 ; \quad a_{n}=3+a_{n-1}$$ Check back soon! Problem 36 A sequence is defined recursively. Write down the first five terms. $$a_{1}=3 ; \quad a_{n}=4-a_{n-1}$$ Check back soon! Problem 37 A sequence is defined recursively. Write down the first five terms. $$a_{1}=-2 ; \quad a_{n}=n+a_{n-1}$$ Check back soon! Problem 38 A sequence is defined recursively. Write down the first five terms. $$a_{1}=1 ; \quad a_{n}=n-a_{n-1}$$ Check back soon! Problem 39 A sequence is defined recursively. Write down the first five terms. $$a_{1}=5 ; \quad a_{n}=2 a_{n-1}$$ Check back soon! Problem 40 A sequence is defined recursively. Write down the first five terms. $$a_{1}=2 ; a_{n}$$ Check back soon! Problem 41 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! Problem 42 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! Problem 42 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! Problem 43 A sequence is defined recursively. Write down the first five terms. $$a_{1}=1 ; \quad a_{2}=2 ; \quad a_{n}=a_{n-1} \cdot a_{n-2}$$ Check back soon! Problem 44 A sequence is defined recursively. Write down the first five terms. $$a_{1}=-1 ; \quad a_{2}=1 ; \quad a_{n}=a_{n-2}+n a_{n-1}$$ Check back soon! Problem 45 A sequence is defined recursively. Write down the first five terms. $$a_{1}=A ; \quad a_{n}=a_{n-1}+d$$ Check back soon! Problem 46 A sequence is defined recursively. Write down the first five terms. $$a_{1}=A ; \quad a_{n}=r a_{n-1}, \quad r \neq 0$$ Check back soon! Problem 47 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! Problem 48 A sequence is defined recursively. Write down the first five terms. $$a_{1}=\sqrt{2} ; \quad a_{n}=\sqrt{\frac{a_{n-1}}{2}}$$ Check back soon! Problem 49 Write out each sum. $$\sum_{k=1}^{n}(k+2)$$ Check back soon! Problem 50 Write out each sum. $$\sum_{i=1}^{n}(2 k+1)$$ Check back soon! Problem 51 Write out each sum. $$\sum_{k=1}^{n} \frac{k^{2}}{2}$$ Check back soon! Problem 52 Write out each sum. $$\sum_{k=1}^{n}(k+1)^{2}$$ Check back soon! Problem 53 Write out each sum. $$\sum_{k=0}^{n} \frac{1}{3^{k}}$$ Check back soon! Problem 54 Write out each sum. $$\hat{\Sigma}\left(\frac{3}{2}\right)^{k}$$ Check back soon! Problem 55 Write out each sum. $$\sum_{k=1}^{n-1} \frac{1}{3^{k+1}}$$ Check back soon! Problem 56 Write out each sum. $$\sum_{k=0}^{n-1}(2 k+1)$$ Check back soon! Problem 57 Write out each sum. $$\sum_{k=2}^{n}(-1)^{k} \ln k$$ Check back soon! Problem 58 Write out each sum. $$\sum_{k=1}^{n}(-1)^{k+1} 2^{k}$$ Check back soon! Problem 59 Express each sum using summation notation. $$1+2+3+\dots+20$$ Check back soon! Problem 60 Express each sum using summation notation. $$1^{3}+2^{3}+3^{3}+\dots+8^{3}$$ Check back soon! Problem 61 Express each sum using summation notation. $$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+\frac{13}{13+1}$$ Check back soon! Problem 62 Express each sum using summation notation. $$1+3+5+7+\dots+[2(12)-1]$$ Check back soon! Problem 63 Express each sum using summation notation. $$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots+(-1)^{6}\left(\frac{1}{3^{6}}\right)$$ Check back soon! Problem 64 Express each sum using summation notation. $$\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\dots+(-1)^{12}\left(\frac{2}{3}\right)^{11}$$ Check back soon! Problem 65 Express each sum using summation notation. $$3+\frac{3^{2}}{2}+\frac{3^{3}}{3}+\dots+\frac{3^{n}}{n}$$ Check back soon! Problem 66 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! Problem 67 Express each sum using summation notation. $$a+(a+d)+(a+2 d)+\cdots+(a+n d)$$ Check back soon! Problem 68 Express each sum using summation notation. $$a+a r+a r^{2}+\dots+a r^{n-1}$$ Check back soon! Problem 69 Find the sum of each sequence. $$\sum_{k=1}^{\infty} 5$$ Check back soon! Problem 70 Find the sum of each sequence. $$\sum_{k=1}^{50} 8$$ Check back soon! Problem 71 Find the sum of each sequence. $$\sum_{k=1}^{\infty} k$$ Check back soon! Problem 72 Find the sum of each sequence. $$\sum_{k=1}^{24}(-k)$$ Check back soon! Problem 73 Find the sum of each sequence. $$\sum_{k=1}^{24}(-k)$$ Check back soon! Problem 74 Find the sum of each sequence. $$\sum_{k=1}^{26}(3 k-7)$$ Check back soon! Problem 75 Find the sum of each sequence. $$\sum_{k=1}^{16}\left(k^{2}+4\right)$$ Check back soon! Problem 76 Find the sum of each sequence. $$\sum_{k=0}^{14}\left(k^{2}-4\right)$$ Check back soon! Problem 77 Find the sum of each sequence. $$\sum_{t=1}^{\infty}(2 k)$$ Check back soon! Problem 78 Find the sum of each sequence. $$\sum_{i}(-3 k)$$ Check back soon! Problem 79 Find the sum of each sequence. $$\sum_{i=1}^{\infty} k^{3}$$ Check back soon! Problem 80 Find the sum of each sequence. $$\sum_{i=1}^{2 i} k^{3}$$ Check back soon! 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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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! 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$p_{2}$. Check back soon! 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) Check back soon! 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. Check back soon! 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) Check back soon! 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. Check back soon! 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 Check back soon! 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. Check back soon! 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. Check back soon! 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. Check back soon! 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. $$\sqrt{5}$$ Check back soon! 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. $$\sqrt{8}$$ Check back soon! 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. $$\sqrt{21}$$ Check back soon! 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. $$\sqrt{89}$$ Check back soon! 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. Check back soon! Problem 98 For the sequence given in Problem$97,$show that $$u_{n+1}=\frac{(n+1)(n+2)}{2}$$. Check back soon! 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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