Chapter 9

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \(\left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{3}\)

Use a graphing utility to find the sum. $$\sum_{j=1}^{6}(24-3 j)$$

Use Gauss-Jordan elimination to solve the system of equations. $$\left\\{\begin{array}{l} 2 x-y+7 z=-10 \\ 3 x+2 y-4 z=17 \\ 6 x-5 y+z=-20 \end{array}\right.$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \(\left(-\frac{1}{3}+\frac{\sqrt{3}}{3} i\right)^{3}\)

Use a graphing utility to find the sum. $$\sum_{j=1}^{10} \frac{6}{3 j+1}$$

Use Gauss-Jordan elimination to solve the system of equations. $$\left\\{\begin{array}{c} -x+4 y+10 z=4 \\ 5 x-3 y+z=31 \\\8 x+2 y-3 z=-5\end{array}\right.$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \(\left(\frac{1}{4}-\frac{\sqrt{3}}{4} i\right)^{3}\)

Use a graphing utility to find the sum. $$\sum_{k=0}^{4} \frac{(-1)^{k}}{(k+1) !}$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \(\left(\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)^{3}\)

Use a graphing utility to find the sum. $$\sum_{k=0}^{4} \frac{(-1)^{k}}{k !}$$

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\cdots+\frac{1}{3(9)}$$

Use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise \(103,\) use the expansion \((1.02)^{8}=(1+0.02)^{8}\) $$ =1+8(0.02)+28(0.02)^{2}+\cdots $$ \((2.005)^{10}\)

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\cdots+\frac{5}{1+15}$$

Use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise \(103,\) use the expansion \((1.02)^{8}=(1+0.02)^{8}\) $$ =1+8(0.02)+28(0.02)^{2}+\cdots $$ \((2.99)^{12}\)

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\left[2\left(\frac{1}{8}\right)+3\right]+\left[2\left(\frac{2}{8}\right)+3\right]+\cdots+\left[2\left(\frac{8}{8}\right)+3\right]$$

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\left[1-\left(\frac{1}{6}\right)^{2}\right]+\left[1-\left(\frac{2}{6}\right)^{2}\right]+\cdots+\left[1-\left(\frac{6}{6}\right)^{2}\right]$$

Physics The temperature of water in an ice cube tray is \(70^{\circ} \mathrm{F}\) when it is placed in a freezer. Its temperature \(n\) hours after being placed in the freezer is \(20 \%\) less than I hour earlier. (a) Find a formula for the \(n\)th term of the geometric sequence that gives the temperature of the water \(n\) hours after it is placed in the freezer. (b) Find the temperatures of the water 6 hours and 12 hours after it is placed in the freezer. (c) Use a graphing utility to graph the sequence to approximate the time required for the water to freeze.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. \(f(x)=x^{4}-5 x^{2}\) \(g(x)=f(x-2)\)

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$-3+9-27+81-243+729$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. \(f(x)=x^{3}-4 x\) \(g(x)=f(x+4)\)

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdot \cdot \cdot-\frac{1}{128}$$

In a fractal, a geometric figure is repeated at smaller and smaller scales. The sphere flake shown is a computer-generated fractal that was created by Eric Haines. The radius of the large sphere is \(1 .\) Attached to the large sphere are nine spheres of radius \(\frac{1}{3} .\) Attached to each of the smaller spheres are nine spheres of radius \(\frac{1}{9} .\) This process is continued infinitely. (a) Write a formula in series notation that gives the surface area of the sphereflake. (b) Write a formula in series notation that gives the volume of the sphereflake. (c) Is the surface area of the sphereflake finite or infinite? Is the volume finite or infinite? If either is finite, find the value.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. \(f(x)=-x^{3}+3 x^{2}-4\) \(g(x)=f(x+5)\)

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\cdots \cdot-\frac{1}{20^{2}}$$

Manufacturing The manufacturer of a new food processor plans to produce and sell 8000 units per year. Each year, \(10 \%\) of all units sold become inoperative. So, 8000 units will be in use after 1 year, \([8000+0.9(8000)]\) units will be in use after 2 years, and so on. (a) Write a formula in series notation for the number of units that will be operative after \(n\) years. (b) Find the numbers of units that will be operative after 10 years, 15 years, and 20 years. (c) If this trend continues indefinitely, will the number of units that will be operative be finite? If so, how many? If not, explain your reasoning.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. \(f(x)=-x^{4}+4 x^{2}-1\) \(g(x)=f(x-3)\)

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{1 \cdot 3}-\frac{1}{2 \cdot 4}+\frac{1}{3 \cdot 5}-\frac{1}{4 \cdot 6}+\cdot \cdot \cdot-\frac{1}{10 \cdot 12}$$

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}$$

Use a graphing utility to graph the functions in the given order and in the same viewing window. Compare the graphs. Which two functions have identical graphs, and why? (a) \(f(x)=\left(1-\frac{1}{2} x\right)^{4}\) (b) \(g(x)=1-2 x+\frac{3}{2} x^{2}\) (c) \(h(x)=1-2 x+\frac{3}{2} x^{2}-\frac{1}{2} x^{3}\) (d) \(p(x)=1-2 x+\frac{3}{2} x^{2}-\frac{1}{2} x^{3}+\frac{1}{16} x^{4}\)

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{2}+\frac{2}{4}+\frac{6}{8}+\frac{24}{16}+\frac{120}{32}+\frac{720}{64}$$

True or False? Determine whether the statement is true or false. Justify your answer. The first \(n\) terms of a geometric sequence with a common ratio of 1 are the same as the first \(n\) terms of an arithmetic sequence with a common difference of 0 if both sequences have the same first term.

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is \(p\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment. A fair coin is tossed seven times. To find the probability of obtaining four heads, evaluate the term $$_{7} C_{4}\left(\frac{1}{2}\right)^{4}\left(\frac{1}{2}\right)^{3}$$ in the expansion of\(\left(\frac{1}{2}+\frac{1}{2}\right)^{7}\).

Find the indicated partial sum of the series. \(\sum_{i=1}^{\infty} \frac{7}{5^{i}}\) Fourth partial sum

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is \(p\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment. The probability of a baseball player getting a hit during any given time at bat is \(\frac{1}{4} .\) To find the probability that the player gets three hits during the next 10 times at bat, evaluate the term $$_{10} C_{3}\left(\frac{1}{4}\right)^{3}\left(\frac{3}{4}\right)^{7}$$ in the expansion of \(\left(\frac{1}{4}+\frac{3}{4}\right)^{10}\).

Find the indicated partial sum of the series. \(\sum_{i=1}^{\infty} \frac{2}{3^{i}}\) Fifth partial sum

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is \(p\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment. The probability of a sales representative making a sale to any one customer is \(\frac{1}{3} .\) The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term $$_{8} C_{4}\left(\frac{1}{3}\right)^{4}\left(\frac{2}{3}\right)^{4}$$ in the expansion of \(\left(\frac{1}{3}+\frac{2}{3}\right)^{8}\).

Find the indicated partial sum of the series. \(\sum_{n=1}^{\infty} 4\left(-\frac{1}{2}\right)^{n}\) Third partial sum

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is \(p\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment. To find the probability that the sales representative in Exercise 115 makes four sales when the probability of a sale to any one customer is \(\frac{1}{2},\) evaluate the term $$_{8} C_{4}\left(\frac{1}{2}\right)^{4}\left(\frac{1}{2}\right)^{4}$$ in the expansion of \(\left(\frac{1}{2}+\frac{1}{2}\right)^{8}\).

Find the indicated partial sum of the series. \(\sum_{n=1}^{\infty} 8\left(-\frac{1}{4}\right)^{n}\) Fourth partial sum

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence. $$a_{1}=100, r=e^{x}, 9 \text { th term }$$

Find (a) the fourth partial sum and (b) the sum of the infinite series. $$\sum_{i=1}^{\infty} \frac{6}{10^{i}}$$

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence. $$a_{1}=4, r=(4 x) / 3,6 \text { th term }$$

The amounts \(f\) (in billions of dollars) of child support collected in the United States from 2000 through 2013 can be approximated by the model $$\begin{aligned} &f(t)=-0.039 t^{2}+1.30 t+17.7\\\ &0 \leq t \leq 13 \end{aligned}$$ where \(t\) represents the year, with \(t=0\) corresponding to 2000 (see figure). (a) Adjust the model so that \(t=0\) corresponds to 2005 rather than \(2000 .\) To do this, shift the graph of \(f\) five units to the left and obtain \(g(t)=f(t+5)\) Write \(g(t)\) in standard form. (b) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window.

Find (a) the fourth partial sum and (b) the sum of the infinite series. $$\sum_{k=1}^{\infty} \frac{4}{10^{k}}$$

Exploration Use a graphing utility to graph each function. Identify the horizontal asymptote of the graph and determine its relationship to the sum. (a) \(f(x)=6\left[\frac{1-(0.5)^{x}}{1-(0.5)}\right], \sum_{n=0}^{\infty} 6\left(\frac{1}{2}\right)^{n}\) (b) \(f(x)=2\left[\frac{1-(0.8)^{x}}{1-(0.8)}\right], \sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n}\)

Find (a) the fourth partial sum and (b) the sum of the infinite series. $$\sum_{k=1}^{\infty} 7\left(\frac{1}{10}\right)^{k}$$

Writing Write a brief paragraph explaining why the terms of a geometric sequence decrease in magnitude when \(-1

Find (a) the fourth partial sum and (b) the sum of the infinite series. $$\sum_{i=1}^{\infty} 2\left(\frac{1}{10}\right)^{i}$$

Write a brief paragraph explaining how to use the first two terms of a geometric sequence to find the \(n\)th term.

In your own words, explain how to form the rows of Pascal's Triangle. Then form rows \(8-10\) of Pascal's Triangle.