Chapter 11

Find the sum of the following series in two ways: by adding terms and by using the geometric series formula. $$ 3+3 \cdot 2+3 \cdot 2^{2} $$

In 2008 , world oil consumption was \(29.3\) billion barrels, \({ }^{6}\) a decrease of \(5.5 \%\) from 2007 . Assuming that consumption continues to decrease at the same percentage rate, make a table showing yearly consumption between 2008 and 2017 , inclusive. Find the total quantity of oil consumed during this decade.

A yearly deposit of \(\$ 1000\) is made into a bank account that pays \(8.5 \%\) interest per year, compounded annually. What is the balance in the account right after the \(20^{\text {th }}\) deposit? How much of the balance comes from the annual deposits and how much comes from interest?

Find the sum of the following series in two ways: by adding terms and by using the geometric series formula. $$ 50+50(0.9)+50(0.9)^{2}+50(0.9)^{3} $$

Find the sum, if it exists. $$ 5+5 \cdot 3+5 \cdot 3^{2}+\cdots+5 \cdot 3^{12} $$

Every morning, a patient receives a \(50-\mathrm{mg}\) injection of a drug. At the end of a 24 -hour period, \(60 \%\) of the drug remains in the body. What quantity of drug is in the body (a) Right after the \(3^{\mathrm{rd}}\) injection? (b) Right after the \(7^{\text {th }}\) injection? (c) Right after an injection, at the steady state?

An annuity earning \(0.5 \%\) per month, compounded monthly, is to make 36 monthly payments of \(\$ 1000\) each. starting now. What is the present value of this annuity?

Find the sum, if it exists. $$ 20+20(1.45)+20(1.45)^{2}+\cdots+20(1.45)^{14} $$

An annuity makes annual payments of \(\$ 50,000\), starting now, from an account paying \(7.2 \%\) interest per year, compounded annually. Find the present value of the annuity if it makes (a) Ten payments (b) Payments in perpetuity

Find the sum, if it exists. $$ 100+100(0.85)+100(0.85)^{2}+\cdots+100(0.85)^{10} $$

A dose of \(120 \mathrm{mg}\) is taken by a patient at the same time every day. In one day, \(30 \%\) of the drug is excreted. (a) At the steady state, find the quantity of drug in the body right after a dose. (b) Check that at the steady state, the quantity excreted in one day is equal to the dose.

Twenty annual payments of \(\$ 5000\) each, with the first payment one year from now, are to be made from an account earning \(10 \%\) per year, compounded annually. How much must be deposited now to cover the payments?

Find the sum, if it exists. $$ 1000+1000(1.05)+1000(1.05)^{2}+\cdots $$

At the same time every day, a patient takes \(50 \mathrm{mg}\) of the antidepressant fluoxetine, whose half-life is 3 days. (a) What fraction of the dose remains in the body after a 24-hour period? (b) What is the quantity of fluoxetine in the body right after taking the \(7^{\text {th }}\) dose? (c) In the long run, what is the quantity of fluoxetine in the body right after a dose?

What is the present value of an annuity that pays \(\$ 20,000\) each year, forever, starting today, from an account that pays \(1 \%\) interest per year, compounded annually?

Find the sum, if it exists. $$ 75+75(0.22)+75(0.22)^{2}+\cdots $$

A person with chronic pain takes a \(30 \mathrm{mg}\) tablet of morphine every 4 hours. The half-life of morphine is 2 hours. (a) How much morphine is in the body right after and right before taking the \(6^{\text {th }}\) tablet? (b) At the steady state, find the quantity of morphine in the body right after and right before taking a tablet.

A deposit of \(\$ 100,000\) is made into an account paying \(8 \%\) interest per year, compounded annually. Annual payments of \(\$ 10,000\) each, starting right after the deposit, are made out of the account. How many payments can be made before the account runs out of money?

Find the sum, if it exists. $$ 500(0.4)+500(0.4)^{2}+500(0.4)^{3}+\cdots $$

(a) An allergy drug with a half-life of 18 weeks is given in \(100-\mathrm{mg}\) doses once a week. At the steady state, find the quantity of the drug in the body right after a dose. (b) The drug does not become effective until the quantity in the body right after a dose reaches \(2000 \mathrm{mg}\). How many weeks after the first dose does the drug become effective?

An employer pays you 1 penny the first day you work and doubles your wages each day after that. Find your total earnings after working 7 days a week for (a) One week (b) Two weeks (c) Three weeks (d) Four weeks

Find the sum, if it exists. $$ 31500+6300+1260+252+\cdots $$

A cigarette puts \(1.2 \mathrm{mg}\) of nicotine into the body. Nicotine leaves the body at a continuous rate of \(34.65 \%\) per hour, but more than \(60 \mathrm{mg}\) can be lethal. If a person smokes a cigarette with each of the following frequencies, find the long-run quantity of nicotine in the body right after a cigarette. Does the nicotine reach the lethal level? (a) Every hour (b) Every half hour (c) Every 15 minutes (d) Every 6 minutes (e) Every 3 minutes

An employee accepts a job with a starting salary of \(\$ 30,000\) and a cost-of- living increase of \(4 \%\) every year for the next 10 years. What is the employee's salary right before the start of the \(11^{\text {th }}\) year? What are her total earnings during the first 10 years?

Find the sum, if it exists. $$ 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^{10}} $$

Each day at lunch a person consumes 8 micrograms of a toxin found in a pesticide; the toxin is metabolized at a continuous rate of \(0.5 \%\) per day. In the long run, how much of this toxin accumulates in the person's body? Give the quantities right after and right before lunch.

Find the sum, if it exists. $$ 65+\frac{65}{1.02}+\frac{65}{(1.02)^{2}}+\cdots+\frac{65}{(1.02)^{18}} $$

At the end of 2008 , the total reserve of a mineral was \(350,000 \mathrm{~m}^{3}\). In the year 2009 , about \(5000 \mathrm{~m}^{3}\) was used. Each year, consumption of the mineral is expected to increase by \(8 \%\). Under these assumptions, in how many years will all reserves of the mineral be depleted?

We use \(1500 \mathrm{~kg}\) of a mineral this year and consumption of the mineral is increasing annually by \(4 \%\). The total reserves of the mineral are estimated to be \(120,000 \mathrm{~kg}\). Approximately when will the reserves run out?

What is the present value of a \(\$ 1000\) bond which pays \(\$ 50\) a year for 10 years, starting one year from now? Assume the interest rate is \(6 \%\) per year, compounded annually.

Find the sum, if it exists. $$ 200+100+50+25+12.5+\cdots $$

At the end of 2007 , natural gas reserves were 180 trillion \(\mathrm{m}^{3}\); during 2007, about 3 trillion \(\mathrm{m}^{3}\) of natural gas were consumed. \({ }^{7}\) Estimate how long natural gas reserves will last if consumption increases at \(2 \%\) per year.

What is the present value of a \(\$ 1000\) bond which pays \(\$ 50\) a year for 10 years, starting one year from now? Assume the interest rate is \(4 \%\) per year, compounded annually.

Find the sum, if it exists. $$ -2+1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\cdots $$

(a) What is the present value of a \(\$ 1000\) bond which pays \(\$ 50\) a year for 10 years, starting one year from now? Assume the interest rate is \(5 \%\) per year, compounded annually. (b) Since \(\$ 50\) is \(5 \%\) of \(\$ 1000\), this bond is called a \(5 \%\) bond. What does your answer to part (a) tell you about the relationship between the principal and the present value of this bond if the interest rate is \(5 \%\) ? (c) If the interest rate is more than \(5 \%\) per year, compounded annually, which is larger: the principal or the present value of the bond? Why is the bond then described as trading at a discount? (d) If the interest rate is less than \(5 \%\) per year, compounded annually, why is the bond described as trading at a premium?

In Example \(3(\mathrm{a})\), we found partial sums of the geometric series with \(a=10\) and \(r=0.75\) and showed that the sum of this series is 40 . Find the partial sums \(S_{n}\) for \(n=5,10,15,20 .\) As \(n\) gets larger, do the partial sums appear to be approaching 40 ?

This problem illustrates how banks create credit and can thereby lend out more money than has been deposited. Suppose that initially \(\$ 100\) is deposited in a bank. Experience has shown bankers that on average only \(8 \%\) of the money deposited is withdrawn by the owner at any time. Consequently, bankers feel free to lend out \(92 \%\) of their deposits. Thus \(\$ 92\) of the original \(\$ 100\) is loaned out to other customers (to start a business, for example). This \(\$ 92\) becomes someone else's income and, sooner or later, is redeposited in the bank. Thus \(92 \%\) of \(\$ 92\), or \(\$ 92(0.92)=\$ 84.64\), is loaned out again and eventually redeposited. Of the \(\$ 84.64\), the bank again loans out \(92 \%\), and so on. (a) Find the total amount of money deposited in the bank as a result of these transactions. (b) The total amount of money deposited divided by the original deposit is called the credit multiplier. Calculate the credit multiplier for this example and explain what this number tells us.

To stimulate the economy in 2002 , the government gave a tax rebate totaling 40 billion dollars. Find the total additional spending resulting from this tax rebate if everyone who receives money spends (a) \(80 \%\) of it (b) \(90 \%\) of it

Every month, $$\$ 500$$ is deposited into an account earning \(0.5 \%\) interest a month, compounded monthly. (a) How much is in the account right after the \(6^{\text {th }}\) deposit? Right before the \(6^{\text {th }}\) deposit? (b) How much is in the account right after the \(12^{\text {th }}\) deposit? Right before the \(12^{\text {th }}\) deposit?

To stimulate the economy in 2008 , the government gave a tax rebate totaling 100 billion dollars. Find the total additional spending resulting from this tax rebate if everyone who receives money spends (a) \(80 \%\) of it (b) \(90 \%\) of it

Each year, a family deposits $$\$ 5000$$ into an account paying \(8.12 \%\) interest per year, compounded annually. How much is in the account right after the \(20^{\text {th }}\) deposit?

(a) A dose \(D\) of a drug is administered at intervals equal to the half-life. (That is, the second dose is given when half the first dose remains.) At the steady state, find the quantity of drug in the body right after a dose. (b) If the quantity of a drug in the body after a dose is \(300 \mathrm{mg}\) at the steady state and if the interval between doses equals the half-life, what is the dose?

Each morning, a patient receives a \(25 \mathrm{mg}\) injection of an anti- inflammatory drug, and \(40 \%\) of the drug remains in the body after 24 hours. Find the quantity in the body: (a) Right after the \(3^{\text {rd }}\) injection. (b) Right after the \(6^{\text {th }}\) injection. (c) In the long run, right after an injection.

A dose, \(D\), of a drug is taken at regular time intervals, and a fraction \(r\) remains after one time interval. Show that at the steady state, the quantity of the drug excreted between doses equals the dose.

A person who deposits money in a bank account starts a long process described by the reserve-deposit ratio, \(r\). For every dollar deposited, the bank keeps \(r\) dollars and lends \(1-r\) dollars to someone else, who deposits the loan in a bank account. The same fraction of the second deposit is loaned out, to be deposited in turn, and so on. If the initial deposit is \(N\) dollars, find the total value of the bank accounts generated by this deposit: (a) After the second deposit (b) After the third deposit (c) If the process continues forever

A smoker inhales \(0.4 \mathrm{mg}\) of nicotine from a cigarette. After one hour, \(71 \%\) of the nicotine remains in the body. If a person smokes one cigarette every hour beginning at 7 \(\mathrm{am}\), how much nicotine is in the body right after the 11 pm cigarette?

Cephalexin is an antibiotic with a half-life in the body of \(0.9\) hours, taken in tablets of \(250 \mathrm{mg}\) every six hours. (a) What percentage of the cephalexin in the body at the start of a six-hour period is still there at the end (assuming no tablets are taken during that time)? (b) Write an expression for \(Q_{1}, Q_{2}, Q_{3}, Q_{4}\), where \(Q_{n}\) \(\mathrm{mg}\) is the amount of cephalexin in the body right after the \(n^{\text {th }}\) tablet is taken. (c) Express \(Q_{3}, Q_{4}\) in closed form and evaluate them. (d) Write an expression for \(Q_{n}\) and put it in closed form. (e) If the patient keeps taking the tablets, use your answer to part (d) to find the quantity of cephalexin in the body in the long run, right after taking a tablet.