Chapter 20

Find the first term of an AP whose common difference is 6 and whose tenth term is 77

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$\left(2 a^{2}+\sqrt{b}\right)^{5}=32 a^{10}+80 a^{8} b^{1 / 2}+80 a^{6} b+40 a^{4} b^{3 / 2}+10 a^{2} b^{2}+b^{5 / 2}$$

Insert three geometric means between 5 and \(1280 .\)

Find the sum of the first 12 terms of the AP $$3,6,9,12, \dots$$

Insert three geometric means between 144 and \(9 .\)

Find the sum of the first five terms of the AP $$1,5,9,13, \dots$$

Verify the first four terms of each binomial expansion. $$\left(x^{2}+y^{3}\right)^{8}=x^{16}+8 x^{14} y^{3}+28 x^{12} y^{6}+56 x^{10} y^{9}+\cdots$$

Exponential Growth: Using the equation for exponential growth, $$y=a e^{n t}$$ with \(a=1\) and \(n=0.5,\) compute values of \(y\) for \(t=0,1,2, \ldots ., 10 .\) Show that while the values of \(t\) form an \(\mathrm{AP}\), the values of \(y\) form a GP. Find the common ratio.

Find the sum of the first nine terms of the AP \(5,10,15,20, \dots\)

Verify the first four terms of each binomial expansion. $$\left(x^{2}-2 y^{3}\right)^{11}=x^{22}-22 x^{20} y^{3}+220 x^{18} y^{6}-1320 x^{16} y^{9}+\cdots$$

Find the sum of the first 20 terms of the AP $$1,3,5,7, \dots$$

Verify the first four terms of each binomial expansion. $$\left(a-b^{4}\right)^{9}=a^{9}-9 a^{8} b^{4}+36 a^{7} b^{8}-84 a^{6} b^{12}+\cdots$$

Cooling: A certain iron casting is at \(1800^{\circ} \mathrm{F}\) and cools so that its temperature at each minute is \(10 \%\) less than its temperature the preceding minute. Find its temperature after 1 h.

How many terms of the \(\mathrm{AP} 4,7,10, \ldots\) will give a sum of \(375 ?\)

Verify the first four terms of each binomial expansion. $$\left(a^{3}+2 b\right)^{12}=a^{36}+24 a^{33} b+264 a^{30} b^{2}+1760 a^{27} b^{3}+\cdots$$

Light Through an Absorbing Medium: Sunlight passes through a glass filter. Each millimeter of glass absorbs \(20 \%\) of the light passing through it. What percentage of the original sunlight will remain after passing through \(5.0 \mathrm{mm}\) of the glass?

How many terms of the AP \(2,9,16, \ldots\) will give a sum of \(270 ?\)

Write the requested term of each binomial expansion, and simplify. Seventh term of \(\left(a^{2}-2 b^{3}\right)^{12}\)

Radioactive Decay: A certain radioactive material decays so that after each year the radioactivity is \(8 \%\) less than at the start of that year. How many years will it take for its radioactivity to be \(50 \%\) of its original value?

Insert two arithmetic means between 5 and \(20 .\)

Write the requested term of each binomial expansion, and simplify. Eleventh term of \((2-x)^{16}\)

Pendulum: Each swing of a certain pendulum is \(85.0 \%\) as long as the one before. If the first swing is 12.0 in., find the entire distance traveled in eight swings.

Insert five arithmetic means between 7 and 25

Write the requested term of each binomial expansion, and simplify. Fourth term of \((2 a-3 b)^{7}\)

Bouncing Ball: A ball dropped from a height of 10.0 ft rebounds to half its height on each bounce. Find the total distance traveled when it hits the ground for the fifth time.

Insert four arithmetic means between -6 and -9

Write the requested term of each binomial expansion, and simplify. Eighth term of \((x+a)^{11}\)

Population Growth: One of the most famous and controversial references to arithmetic and geometric progressions was made by Thomas Malthus in \(1798 .\) He wrote: "Population, when unchecked, increases in a geometrical ratio, and subsistence for man in an arithmetical ratio." Each day the size of a certain colony of bacteria is \(25 \%\) larger than on the preceding day. If the original size of the colony was 10,000 bacteria, find its size after 5 days.

Insert three arithmetic means between 20 and 56

Write the requested term of each binomial expansion, and simplify. Fifth term of \((x-2 \sqrt{y})^{25}\)

A person has two parents, and each parent has two parents, and so on. We can write a GP for the number of ancestors as \(2,4,8, \ldots .\) Find the total number of ancestors in five generations, starting with the parents' generation.

Find the fourth term of the harmonic progression $$\frac{3}{5}, \frac{3}{8}, \frac{3}{11}, \dots$$

Write the requested term of each binomial expansion, and simplify. Ninth term of \(\left(x^{2}+1\right)^{15}\)

Musical Scale: The frequency of the "A" note above middle C is, by international agreement, equal to \(440 \mathrm{Hz}\). A note one octave higher is at twice that frequency, or \(880 \mathrm{Hz}\). The octave is subdivided into 12 half-tone intervals, where cach half-tone is higher than the one preceding by a factor equal to the twelfth root of \(2 .\) This is called the equally tempered scale and is usually attributed to Johann Sebastian Bach \((1685-1750) .\) Write a GP showing the frequency of each half-tone, from 440 to 880 Hz. Work to two decimal places.

Verify the first four terms of each infinite binomial series. $$(1-a)^{2 / 3}=1-2 a / 3-a^{2} / 9-4 a^{3} / 81 \ldots$$

Chemical Reactions: Increased temperature usually causes chemicals to react faster. If a certain reaction proceeds \(15 \%\) faster for each \(10^{\circ} \mathrm{C}\) increase in temperature, by what factor is the reaction speed increased when the temperature rises by \(50^{\circ} \mathrm{C} ?\)

Show that the harmonic mean between two numbers \(a\) and \(b\) is given by $$\text { Harmonic Mean }=\frac{2 a b}{a+b}$$

Mixtures: A radiator contains \(30 \%\) antifreeze and \(70 \%\) water. One-fourth of the mixture is removed and replaced by pure water. If this procedure is repeated three more times, find the percent antifreeze in the final mixture.

Insert two harmonic means between \(\frac{7}{9}\) and \(\frac{7}{15}\)

Verify the first four terms of each infinite binomial series. $$(1+5 a)^{-5}=1-25 a+375 a^{2}-4375 a^{3} \dots$$

Energy Consumption: If the U.S. energy consumption is \(7.00 \%\) higher each year, by what factor will the energy consumption have increased after 10.0 years?

Insert three harmonic means between \(\frac{6}{21}\) and \(\frac{6}{5}\)

Verify the first four terms of each infinite binomial series. $$(1+a)^{-3}=1-3 a+6 a^{2}-10 a^{3} \dots$$

Atmospheric Pressure: The pressures measured at 1 -mi intervals above sea level form a GP, with each value smaller than the preceding by a factor of \(0.819 .\) If the pressure at sea level is 29.92 in. Hg, find the pressure at an altitude of 5 mi.

Verify the first four terms of each infinite binomial series. $$1 / \sqrt[6]{1-a}=1+a / 6+7 a^{2} / 72+91 a^{3} / 1296 \ldots$$

Compound Interest: A person deposits \(\$ 10,000\) in a bank giving \(6 \%\) interest, compounded annually. Find to the nearest dollar the value of the deposit after 50 years.

Inflation: The price of a certain house, now \(\$ 126,000,\) is expected to increase by \(5 \%\) each year. Write a GP whose terms are the value of the house at the end of each year, and find the value of the house after 5 years.

Straight-Line Depreciation: A certain milling machine has an initial value of \(\$ 150,000\) and a scrap value of \(\$ 10,000\) twenty years later. Assuming that the machine depreciates the same amount each year, find its value after 8 years. To find the amount of depreciation for each year, divide the total depreciation (initial value - scrap value) by the number of years of depreciation.

Freely Falling Body: A freely falling body falls \(g / 2\) feet during the first second, \(3 g / 2\) feet during the next second, \(5 g / 2\) feet during the third second, and so on, where \(g \approx 32.2 \mathrm{ft} / \mathrm{s}^{2} .\) Find the total distance the body falls during the first \(10 \mathrm{s}\)

Computer: Write a program or use a spreadsheet to generate the terms of a series, given the general term or a recursion relation. Have the program compute and print each term, the partial sum, and the ratio of that term to the preceding one. Use the program to determine if each of the following series converges or diverges. (a) \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}+\cdots\) (b) \(\frac{3}{2}+\frac{9}{8}+\frac{27}{24}+\frac{81}{64}+\cdots+\frac{3^{n}}{n \cdot 2^{n}}+\cdots\) (c) \(1+\frac{4}{7}+\frac{9}{49}+\frac{16}{343}+\dots+\frac{n^{2}}{7^{n-1}}+\cdots\)