Chapter 13

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 \(a_{m}\) Insert three geometric means between 5 and 80 .

Find a formula for the nth 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 \(2 . ]\)

Find the sum of the first ten terms of the sequence $$ a+b, a^{2}+2 b, a^{3}+3 b, a^{4}+4 b, \ldots $$

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 \(F_{m}\)

Compound Interest Julio deposits \(\$ 2000\) in a savings account that pays 2.4\(\%\) interest per year compounded monthly. The amount in the account after \(n\) months is given by the sequence $$A_{n}=2000\left(1+\frac{0.024}{12}\right)^{n}$$ (a) Find the first six terms of the sequence. (b) Find the amount in the account after 3 years.

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? Graph cannot copy

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.

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 \(n\) th bounce.

Population of a City \(A\) 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 .

Paying off a Debt 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 nth 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.

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? \(n\) times?

Fish Farming 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?

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 C that is an octave higher has a frequency of \(512 .\) Find the frequency of \(\mathrm{C}\) two octaves below middle \(\mathrm{C}\) .

Price of a House 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 \(P_{m}\) (b) Find the expected median price in 2010 .

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.

Salary Increases 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 \(S_{m}\) (b) Find his salary in his fifth year of employment.

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?

Concentration of a Solution 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 \(\mathrm{g} / \mathrm{L}\) 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 \(C_{m}\) (b) Find the salt concentration after 8 days.

Drug Concentration 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 nth 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.

Different Sequences That Start the Same (a) Show that the first four terms of the sequence \(a_{n}=n^{2}\) are $$1,4,9,16, \ldots$$ (b) Show that the first four terms of the sequence \(a_{n}=n^{2}+(n-1)(n-2)(n-3)(n-4)\) are also $$1,4,9,16, \ldots$$ (c) Find a sequence whose first six terms are the same as those of \(a_{n}=n^{2}\) but whose succeeding terms differ from this sequence. (d) Find two different sequences that begin $$2,4,8,16, \ldots$$

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.

A Recursively Defined Sequence Find the first 40 terms of the sequence defined by $$a_{n+1}=\left\\{\begin{array}{ll}{\frac{a_{n}}{2}} & {\text { if } a_{n} \text { is an even number }} \\ {3 a_{n}+1} & {\text { if } a_{n} \text { is an odd number }}\end{array}\right.$$ and \(a_{1}=11 .\) Do the same if \(a_{1}=25 .\) Make a conjecture about this type of sequence. Try several other values for \(a_{1},\) to test your conjecture.

A Different Type of Recursion Find the first 10 terms of the sequence defined by $$a_{n}=a_{n-a_{n-1}}+a_{n-a_{n-2}}$$ with $$a_{1}=1 \quad \text { and } \quad a_{2}=1$$ How is this recursive sequence different from the others in this section?

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.

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, as in figure (c). Assuming that this process can be repeated indefinitely, find the total area of all the disks.

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 that is colored blue?

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. $$ \begin{array}{ll}{\text { (a) } 5,-3,5,-3, \ldots} & {\text { (b) } \frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \ldots} \\ {\text { (c) } \sqrt{3}, 3,3 \sqrt{3}, 9,9, \ldots} & {\text { (d) } 1,-1,1,-1, \ldots} \\ {\text { (e) } 2,-1, \frac{1}{2}, 2, \ldots} & {\text { (f) } x-1, x+1, x+2, \ldots} \\\ {\text { (g) }-3,-\frac{3}{2}, 0, \frac{3}{2}, \ldots} & {\text { (h) } \sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1, \ldots}\end{array} $$

Reciprocals of a Geometric Sequence If \(a_{1}, a_{2}\) \(a_{3}, \ldots\) is a geometric sequence with common ratio \(r,\) show that the sequence $$ \frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \ldots $$ is also a geometric sequence, and find the common ratio.

Logarithms of a Geometric Sequence If \(a_{1}, a_{2},\) \(a_{3}, \ldots\) is a geometric sequence with a common ratio \(r>0\) and \(a_{1}>0,\) show that the sequence $$ \log a_{1}, \log a_{2}, \log a_{3}, \ldots $$ is an arithmetic sequence, and find the common difference.

Exponentials of an Arithmetic Sequence If \(a_{1}, a_{2},\) \(a_{3}, \ldots\) is an arithmetic sequence with common difference \(d,\) show that the sequence $$ 10^{a_{1}}, 10^{a_{2}}, 10^{a_{3}}, \ldots $$ is a geometric sequence, and find the common ratio.