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nature of mathematics

Questions and Answers of Nature Of Mathematics

Find the sum of the first \(n\) odd positive integers.
Find the sum of the first \(n\) even positive integers.
Find the sum of the first \(n\) positive integers.
Find the sum of the first 20 terms of the arithmetic sequence whose first term is 100 and whose common difference is 50 .
Find the sum of the first 50 terms of the arithmetic sequence whose first term is -15 and whose common difference is 5 .
Find the sum of the even integers between 41 and 99.
Find the sum of the odd integers between 48 and 136 .
The game of pool uses 15 balls numbered from 1 to 15 (see Figure 11.6). In the game of rotation, a player attempts to "sink" a ball in a pocket of the table and receives the number of points on the
The game of pool uses 15 balls numbered from 1 to 15 (see Figure 11.6). In the game of rotation, a player attempts to "sink" a ball in a pocket of the table and receives the number of points on the
The game of pool uses 15 balls numbered from 1 to 15 (see Figure 11.6). In the game of rotation, a player attempts to "sink" a ball in a pocket of the table and receives the number of points on the
The game of pool uses 15 balls numbered from 1 to 15 (see Figure 11.6). In the game of rotation, a player attempts to "sink" a ball in a pocket of the table and receives the number of points on the
The Peanuts cartoon (page 535) expresses a common feeling regarding chain letters. Consider the total number of letters sent after a particular mailing:\[\begin{array}{lr}\text { 1st mailing: } & 6
In 1935 the first successful chain letters asked a person to "send a dime" to the person at the top of a list of six names, cross off the top name, and then to add their name to the bottom of the
How many blocks would be needed to build a stack like the one shown in Figure 11.7 if the bottom row has 28 blocks? FIGURE 11.7 How many blocks?
Repeat Problem 47 if the bottom row has 87 blocks.Data from Problem 47How many blocks would be needed to build a stack like the one shown in Figure 117 if the bottom row has 28 blocks? FIGURE 11.7
A pendulum is swung \(20 \mathrm{~cm}\) and allowed to swing freely until it eventually comes to rest. Each subsequent swing of the bob of the pendulum is \(90 \%\) as far as the preceding swing. How
The initial swing of the tip of a pendulum is \(25 \mathrm{~cm}\). If each swing of the tip is \(75 \%\) of the preceding swing, how far does the tip travel before eventually coming to rest?
A flywheel is brought to a speed of 375 revolutions per minute (rpm) and allowed to slow and eventually come to rest. If, in slowing, it rotates three-fourths as fast each subsequent minute, how many
A rotating flywheel is allowed to slow to a stop from a speed of \(500 \mathrm{rpm}\). While slowing, each minute it rotates two-thirds as many times as in the preceding minute. How many revolutions
Advertisements say that a new type of superball will rebound to \(9 / 10\) of its original height. If it is dropped from a height of \(10 \mathrm{ft}\), how far, based on the advertisements, will the
A tennis ball is dropped from a height of \(10 \mathrm{ft}\). If the ball rebounds \(2 / 3\) of its height on each bounce, how far will the ball travel before coming to rest?
A culture of bacteria increases by \(100 \%\) every 24 hours. If the original culture contains 1 million bacteria ( \(a_{0}=1\) million), find the number of bacteria present after 10 days.
Use Problem 55 to find a formula for the number of bacteria present after \(d\) days.Data from Problem 55A culture of bacteria increases by 100% every 24 hours. If the original culture contains 1
How many games are necessary for a two-team elimination tournament with 32 teams?
How many games are necessary for a two-team elimination tournament with 128 teams?
Games like Wheel of Fortune and Jeopardy have one winner and two losers. A three-team game tournament is illustrated by Figure 11.8. If Jeopardy has a Tournament of Champions consisting of 27
How many games are necessary for a three-team elimination tournament with 729 teams?
What do we mean by a lump-sum problem?
Why should we call an annuity a periodic payment problem?
What is an annuity problem?
What is a sinking fund problem?
Distinguish an annuity problem from a sinking fund problem.
Describe Example 2 and comment on its possible relevance.Example 2Suppose you are 21 years old and will make monthly deposits to a bank account paying 4% annual interest compounded monthly. Option 1:
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
Use a calculator to evaluate an ordinary annuity formula\[A=m\left[\frac{\left(1+\frac{r}{n}ight)^{n t}-1}{\frac{r}{n}}ight]\]for \(m, r\), and \(t\) (respectively) given in Problems 7-22. Assume
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 23.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 24.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 25.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 26.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 28.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 29.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 30.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 31.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 32.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 33.
In Problems 23-34, find the value of each annuity at the end of the indicated number of years. Assume that the interest is compounded with the same frequency as the deposits. Amount of Deposit m 34.
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed A 35. Frequency n $7,000 annually Rate r 0.5% Time t 5 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed A 36. Frequency n $25,000 annually Rate r 1% Time t 5 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed A $25,000 semiannually 37. Frequency n Rate r 2% Time t 5 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed Frequency n $50,000 semiannually 38. A Rate r 4% Time t 10 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed Frequency n A 39. $165,000 semiannually Rate r 2% Time t 10 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed Frequency A n 40. $3,000,000 semiannually Rate r 3% Time t 20 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed Frequency n A 41. $500,000 quarterly Rate r 8% Time t 10 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed A $55,000 quarterly Frequency n 42. Rate r 1.2% Time t 5 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed Frequency n A 43. $100,000 quarterly Rate r 8% Time t 8
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed 44. Frequency n A $35,000 quarterly Rate r 8% Time t 12 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed 45. Frequency n A $45,000 monthly Rate r 7% | Time t 30 yr
Find the amount of periodic payment necessary for each deposit to a sinking fund in Problems 35-46. Amount Needed Frequency n A 46. $120,000 quarterly Rate r 7% Time t 30 yr
Self-employed persons can make contributions for their retirement into a special tax-deferred account called a Keogh account. Suppose you are able to contribute \(\$ 20,000\) into this account at the
The owner of Sebastopol Tree Farm deposits \(\$ 650\) at the end of each quarter into an account paying \(1.75 \%\) compounded quarterly. What is the value of the account at the end of 5 years?
The owner of Oak Hill Squirrel Farm deposits \(\$ 1,000\) at the end of each quarter into an account paying 1.5\% compounded quarterly. What is the value at the end of 5 years, 6 months?
Clearlake Optical has a \(\$ 50,000\) note that comes due in 4 years. The owners wish to create a sinking fund to pay this note. If the fund earns \(2.5 \%\) compounded semiannually, how much must
A business must raise \(\$ 70,000\) in 5 years. What should be the size of the owners' quarterly payment to a sinking fund paying \(3 \%\) compounded quarterly?
A lottery offers a \(\$ 1,000,000\) prize to be paid in 20 equal installments of \(\$ 50,000\) at the end of each year. What is the future value of this annuity if the current annual rate is \(5 \%\)
A lottery offers a \(\$ 1,000,000\) prize to be paid in 29 equal annual installments of \(\$ 20,000\) with a 30th final payment of \(\$ 420,000\). What is the total value of this annuity if the
John and Rosamond want to retire in 5 years and can save \(\$ 150\) every three months. They plan to deposit the money at the end of each quarter into an account paying \(6.72 \%\) compounded
In 2012 the maximum Social Security deposit by an individual was \(\$ 8,386.75\). Suppose you are 25 and make a deposit of this amount into an account at the end of each year. How much would you have
You want to retire at age 65 . You decide to make a deposit to yourself at the end of each year into an account paying 3\%, compounded annually. Assuming you are now 25 and can spare \(\$ 1,200\) per
Repeat Problem 56 using your own age.Data from Problem 56You want to retire at age 65 . You decide to make a deposit to yourself at the end of each year into an account paying 3\%, compounded
Repeat Problem 55 using your own age.Data from Problem 55In 2012 the maximum Social Security deposit by an individual was \(\$ 8,386.75\). Suppose you are 25 and make a deposit of this amount into an
Clearlake Optical has developed a new lens. The owners plan to issue a \(\$ 4,000,00030\)-year bond with a contract rate of \(5.5 \%\) paid annually to raise capital to market this new lens. This
The owners of Bardoza Greeting Cards wish to introduce a new line of cards but need to raise \(\$ 200,000\) to do it. They decide to issue 10 -year bonds with a contract rate of \(6 \%\) paid
What does amortization mean?
Describe when you would use the present value of an annuity formula.
Distinguish between simple addon interest and amortization.
The variables \(m, n, r, t, A\), and \(P\) are used in the various financial formulas. Tell what each of these variables represents.
Use a calculator to evaluate the present value of an annuity formula\[P=m\left[\frac{1-\left(1+\frac{r}{n}ight)^{-n t}}{\frac{r}{n}}ight]\]for the values of the variables \(m, r\), and \(t\)
Use a calculator to evaluate the present value of an annuity formula\[P=m\left[\frac{1-\left(1+\frac{r}{n}ight)^{-n t}}{\frac{r}{n}}ight]\]for the values of the variables \(m, r\), and \(t\)
Use a calculator to evaluate the present value of an annuity formula\[P=m\left[\frac{1-\left(1+\frac{r}{n}ight)^{-n t}}{\frac{r}{n}}ight]\]for the values of the variables \(m, r\), and \(t\)
Use a calculator to evaluate the present value of an annuity formula\[P=m\left[\frac{1-\left(1+\frac{r}{n}ight)^{-n t}}{\frac{r}{n}}ight]\]for the values of the variables \(m, r\), and \(t\)
Use a calculator to evaluate the present value of an annuity formula\[P=m\left[\frac{1-\left(1+\frac{r}{n}ight)^{-n t}}{\frac{r}{n}}ight]\]for the values of the variables \(m, r\), and \(t\)
Use a calculator to evaluate the present value of an annuity formula\[P=m\left[\frac{1-\left(1+\frac{r}{n}ight)^{-n t}}{\frac{r}{n}}ight]\]for the values of the variables \(m, r\), and \(t\)
Use a calculator to evaluate the present value of an annuity formula\[P=m\left[\frac{1-\left(1+\frac{r}{n}ight)^{-n t}}{\frac{r}{n}}ight]\]for the values of the variables \(m, r\), and \(t\)
Use a calculator to evaluate the present value of an annuity formula\[P=m\left[\frac{1-\left(1+\frac{r}{n}ight)^{-n t}}{\frac{r}{n}}ight]\]for the values of the variables \(m, r\), and \(t\)
Use a calculator to evaluate the amortization formula\[m=\frac{P\left(\frac{r}{n}ight)}{1-\left(1+\frac{r}{n}ight)^{-n t}}\]for the values of the variables \(P, r\), and \(t\) (respectively) given in

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