Chapter 12

Express the repeating decimal as a fraction. $$ 0.2 \overline{53} $$

Depreciation The purchase value of an office computer is \(\$ 12,500\) . Its annual depreciation is \(\$ 1875 .\) Find the value of the computer after 6 years.

Write the sum without using sigma notation. $$\sum_{k=6}^{9} k(k+3)$$

Express the repeating decimal as a fraction. $$ 0.030303 \ldots $$

An arithmetic sequence has first term \(a_{1}=1\) and fourth term \(a_{4}=16 .\) How many terms of this sequence must be added to get 2356\(?\)

Write the sum without using sigma notation. $$\sum_{k=3}^{100} x^{k}$$

Express the repeating decimal as a fraction. $$ 2.11 \overline{25} $$

Poles in a Pile Telephone poles are stored in a pile with 25 poles in the first layer, 24 in the second, and so on. If there are 12 layers, how many telephone poles does the pile contain?

Write the sum without using sigma notation. $$\sum_{j=1}^{n}(-1)^{j+1} x^{j}$$

Express the repeating decimal as a fraction. $$ 0 . \overline{112} $$

Salary lncreases A man gets a job with a salary of \(\$ 30,000\) a year. He is promised a \(\$ 2300\) raise each subsequent year. Find his total earnings for a 10-year period.

Write the sum using sigma notation. \(1+2+3+4+\cdots+100\)

Express the repeating decimal as a fraction. $$ 0.123123123 \ldots $$

Drive-ln Theater A drive-in theater has spaces for 20 cars in the first parking row, 22 in the second, 24 in the third, and so on. If there are 21 rows in the theater, find the number of cars that can be parked.

Write the sum using sigma notation. \(2+4+6+\cdots+20\)

If the numbers \(a_{1}, a_{2}, \ldots, a_{n}\) form a geometric sequence, then \(a_{2}, a_{3}, \ldots, a_{n-1}\) are geometric means between \(a_{1}\) and \(a_{n} .\) Insert three geometric means between 5 and \(80 .\)

Theater Seating An architect designs a theater with 15 seats in the first row, 18 in the second, 21 in the third, and so on. If the theater is to have a seating capacity of \(870,\) how many rows must the architect use in his design?

Write the sum using sigma notation. \(1^{2}+2^{2}+3^{2}+\dots+10^{2}\)

Falling Ball When an object is allowed to fall freely near the surface of the earth, the gravitational pull is such that the object falls 16 \(\mathrm{ft}\) in the first second, 48 \(\mathrm{ft}\) in the next second, 80 \(\mathrm{ft}\) in the next second, and so on. (a) Find the total distance a ball falls in 6 \(\mathrm{s}\) . (b) Find a formula for the total distance a ball falls in \(n\) seconds.

Write the sum using sigma notation. $$\frac{1}{2 \ln 2}-\frac{1}{3 \ln 3}+\frac{1}{4 \ln 4}-\frac{1}{5 \ln 5}+\cdots+\frac{1}{100 \ln 100}$$

Depreciation A construction company purchases a bulldozer for \(\$ 160,000\) . Each year the value of the bulldozer depreciates by 20\(\%\) of its value in the preceding year. Let \(V_{n}\) be the value of the bulldozer in the \(n\) th year. (Let \(n=1\) be the year the bulldozer is purchased.) (a) Find a formula for \(V_{n}\) . (b) In what year will the value of the bulldozer be less than \(\$ 100,000 ?\)

The Twelve Days of Christmas In the well-known song "The Twelve Days of Christmas," a person gives his sweetheart \(k\) gifts on the \(k\)th day for each of the 12 days of Christmas. The person also repeats each gift identically on each subsequent day. Thus, on the 12th day the sweetheart receives a gift for the first day, 2 gifts for the second, 3 gifts received on the 12th day is a partial sum of an arithmetic sequence. Find this sum.

Write the sum using sigma notation. $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{999 \cdot 1000}$$

Family Tree A person has two parents, four grandparents, eight great- grandparents, and so on. How many ancestors does a person have 15 generations back?

Arithmetic Means The arithmetic mean (or average) of two numbers \(a\) and \(b\) is $$m=\frac{a+b}{2}$$ Note that \(m\) is the same distance from \(a\) as from \(b,\) so \(a, m, b\) is an arithmetic sequence. In general, if \(m_{1}, m_{2}, \ldots, m_{k}\) are equally spaced between \(a\) and \(b\) so that $$a, m_{1}, m_{2}, \ldots, m_{k}, b$$ is an arithmetic sequence, then \(m_{1}, m_{2}, \ldots, m_{k}\) are called \(k\) arithmetic means between \(a\) and \(b\) . (a) Insert two arithmetic means between 10 and 18 . (b) Insert three arithmetic means between 10 and 18 . (c) Suppose a doctor needs to increase a patient's dosage of a certain medicine from 100 \(\mathrm{mg}\) to 300 \(\mathrm{mg}\) per day in five equal steps. How many arithmetic means must be inserted between 100 and 300 to give the progression of daily doses, and what are these means?

Write the sum using sigma notation. $$\frac{\sqrt{1}}{1^{2}}+\frac{\sqrt{2}}{2^{2}}+\frac{\sqrt{3}}{3^{2}}+\cdots+\frac{\sqrt{n}}{n^{2}}$$

Bouncing Ball A ball is dropped from a height of 80 \(\mathrm{ft} .\) The elasticity of this ball is such that it rebounds three-fourths of the distance it has fallen. How high does the ball rebound on the fifth bounce? Find a formula for how high the ball rebounds on the \(n\) the bounce.

Write the sum using sigma notation. $$1+x+x^{2}+x^{3}+\cdots+x^{100}$$

Bacteria Culture A culture initially has 5000 bacteria, and its size increases by 8\(\%\) every hour. How many bacteria are present at the end of 5 hours? Find a formula for the number of bacteria present after \(n\) hours.

Write the sum using sigma notation. \(1-2 x+3 x^{2}-4 x^{3}+5 x^{4}+\cdots-100 x^{99}\)

Mixing Coolant A truck radiator holds 5 gal and is filled with water. A gallon of water is removed from the radiator and replaced with a gallon of antifreeze; then, a gallon of the mixture is removed from the radiator and again replaced by a gallon of antifreeze. This process is repeated indefinitely. How much water remains in the tank after this process is repeated 3 times? 5 times?

Find a formula for the \(n\) th term of the sequence $$\sqrt{2}, \quad \sqrt{2 \sqrt{2}}, \quad \sqrt{2 \sqrt{2 \sqrt{2}}}, \quad \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2}}}, \ldots}$$ [Hint: Write each term as a power of \(2 . ]\)

Musical Frequencies The frequencies of musical notes (measured in cycles per second) form a geometric sequence. Middle \(\mathrm{C}\) has a frequency of \(256,\) and the \(\mathrm{C}\) that is an octave higher has a frequency of \(512 .\) Find the frequency of \(\mathrm{C}\) two octaves below middle \(\mathrm{C}\) .

Define the sequence $$G_{n}=\frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right)$$ Use the \([\text { TABLE }]\) command on a graphing calculator to find the first 10 terms of this sequence. Compare to the Fibonacci sequence \(F_{n}\).

Bouncing Ball A ball is dropped from a height of 9 \(\mathrm{ft.}\) The elasticity of the ball is such that it always bounces up one-third the distance it has fallen. (a) Find the total distance the ball has traveled at the instant it hits the ground the fifth time. (b) Find a formula for the total distance the ball has traveled at the instant it hits the ground the \(n\)th time.

Geometric Savings Plan A very patient woman wishes to become a billionaire. She decides to follow a simple scheme: She puts aside 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on, doubling the number of cents each day. How much money will she have at the end of 30 days? How many days will it take this woman to realize her wish?

Compound Interest Helen deposits \(\$ 100\) at the end of each month into an account that pays \(6 \%\) interest per year compounded monthly. The amount of interest she has accumulated after \(n\) months is given by the sequence $$ I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right) $$ (a) Find the first six terms of the sequence. (b) Find the interest she has accumulated after 5 years.

A city was incorporated in 2004 with a population of 35,000. It is expected that the population will increase at a rate of 2% per year. The population \(n\) years after 2004 is given by the sequence $$P_{n}=35,000(1.02)^{n}$$ (a) Find the first five terms of the sequence. (b) Find the population in 2014.

Drug Concentration \(\quad\) A certain drug is administered once a day. The concentration of the drug in the patient's blood-stream increases rapidly at first, but each successive dose has less effect than the preceding one. The total amount of the drug (in mg) in the bloodstream after the \(n\) th dose is given by $$\sum_{k=1}^{n} 50\left(\frac{1}{2}\right)^{k-1}$$ (a) Find the amount of the drug in the bloodstream after \(n=10\) days. (b) If the drug is taken on a long-term basis, the amount in the bloodstream is approximated by the infinite series \(\sum_{k=1}^{\infty} 50\left(\frac{1}{2}\right)^{k-1} .\) Find the sum of this series.

Margarita borrows \(\$ 10,000\) from her uncle and agrees to repay it in monthly installments of \(\$ 200\) . Her uncle charges 0.5\(\%\) interest per month on the balance. (a) Show that her balance \(A_{n}\) in the \(n\) th month is given recursively by \(A_{0}=10,000\) and $$A_{n}=1.005 A_{n-1}-200$$ (b) Find her balance after six months.

Bouncing Ball A certain ball rebounds to half the height from which it is dropped. Use an infinite geometric series to approximate the total distance the ball travels, after being dropped from 1 \(\mathrm{m}\) above the ground, until it comes to rest.

The median price of a house in Orange County increases by about 6\(\%\) per year. In 2002 the median price was \(\$ 240,000 .\) Let \(P_{n}\) be the median price \(n\) years after 2002. (a) Find a formula for the sequence \(P_{n}\). (b) Find the expected median price in 2010.

A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month, and the farmer harvests 300 catfish per month. (a) Show that the catfish population \(P_{n}\) after \(n\) months is given recursively by \(P_{0}=5000\) and $$P_{n}=1.08 P_{n-1}-300$$ (b) How many fish are in the pond after 12 months?

Geometry The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated for each new square. (See the figure.) (a) Find the sum of the areas of all the squares. (b) Find the sum of the perimeters of all the squares.

A newly hired salesman is promised a beginning salary of \(\$ 30,000\) a year with a \(\$ 2000\) raise every year. Let \(S_{n}\) be his salary in his \(n\) th year of employment. (a) Find a recursive definition of \(S_{n}\) (b) Find his salary in his fifth year of employment.

Geometry A circular disk of radius \(R\) is cut out of paper, as shown in figure (a). Two disks of radius \(\frac{1}{2} R\) are cut out of paper and placed on top of the first disk, as in figure (b), and then four disks of radius \(\frac{1}{4} R\) are placed on these two disks (figure (c)). Assuming that this process can be repeated indefinitely, find the total area of all the disks.

A biologist is trying to find the optimal salt concentration for the growth of a certain species of mollusk. She begins with a brine solution that has 4 g/L of salt and increases the concentration by 10\(\%\) every day. Let \(C_{0}\) denote the initial concentration and \(C_{n}\) the concentration after \(n\) days. (a) Find a recursive definition of \(C_{n}\). (b) Find the salt concentration after 8 days.

Geometry A yellow square of side 1 is divided into nine smaller squares, and the middle square is colored blue as shown in the figure. Each of the smaller yellow squares is in turn divided into nine squares, and each middle square is colored blue. If this process is continued indefinitely, what is the total area colored blue?

Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair that becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is \(F_{n},\) where \(F_{n}\) is the \(n\) th term of the Fibonacci sequence.

Arithmetic or Geometric? The first four terms of a sequence are given. Determine whether these terms can be the terms of an arithmetic sequence, a geometric sequence, or neither. Find the next term if the sequence is arithmetic or geometric. (a) \(5,-3,5,-3, \ldots \quad\) (b) \(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \ldots\) (c) \(\sqrt{3}, 3,3 \sqrt{3}, 9, \ldots\) (d) \(1,-1,1,-1, \ldots\) \(\begin{array}{ll}{\text { (e) } 2,-1, \frac{1}{2}, 2, \ldots} & {\text { (f) } x-1, x, x+1, x+2, \ldots} \\ {\text { (g) }-3,-\frac{3}{2}, 0, \frac{3}{2}, \ldots} & {\text { (h) } \sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1, \ldots}\end{array}\)