Chapter 13

Write the first four terms of the infinite sequence whose nth term is given. \(a_{n}=\frac{(-1)^{2 n}}{n^{2}}\)

Write the first four terms of the infinite sequence whose nth term is given. \(a_{n}=(-1)^{2 n+1} 2^{n-1}\)

Write each series in summation notation. Use the index is and let i begin at I in each summation. $$\ln (2)+\ln (3)+\ln (4)$$

Write out the first four terms in the expansion of each binomial. $$\left(x^{2}+5\right)^{9}$$

Write a formula for the general term of each infinite sequence. \(1,3,5,7,9, \dots\)

Write a formula for the general term of each infinite sequence. \(5,7,9,11,13, \ldots\)

Write a formula for the general term of each infinite sequence. \(1,-1,1,-1, \ldots\)

Write a formula for the general term of each infinite sequence. \(-1,1,-1,1, \ldots\)

Write each series in summation notation. Use the index is and let i begin at I in each summation. $$x_{3}+x_{4}+x_{5}+\cdots+x_{50}$$

Write a formula for the general term of each infinite sequence. \(0,2,4,6,8, \dots\)

Write each series in summation notation. Use the index is and let i begin at I in each summation. $$y_{1}+y_{2}+y_{3}+\cdots+y_{30}$$

Write out the first four terms in the expansion of each binomial. $$\left(\frac{a}{2}+\frac{b}{5}\right)^{8}$$

Write a formula for the general term of each infinite sequence. \(4,6,8,10,12, \dots\)

Write each series in summation notation. Use the index is and let i begin at I in each summation. $$w_{1}+w_{2}+w_{3}+\dots+w_{n}$$

Write a formula for the general term of each infinite sequence. \(3,6,9,12, \dots\)

Write a formula for the general term of each infinite sequence. \(4,8,12,16, \dots\)

Write out the terms of each series. $$\sum_{i=1}^{6} x^{i}$$

Write a formula for the general term of each infinite sequence. \(4,7,10,13, \dots\)

Write a formula for the general term of each infinite sequence. \(3,7,11,15, \dots\)

Write a formula for the general term of each infinite sequence. \(1,-3,9,-27, \dots\)

Write out the terms of each series. $$\sum_{i=1}^{3} i x^{i}$$

Write a formula for the general term of each infinite sequence. \(0,1,4,9,16, \dots\)

Write out the terms of each series. $$\sum_{i=1}^{5} \frac{x}{i}$$

Write a formula for the general term of each infinite sequence. \(0,1,8,27,64, \dots\)

A football is on the 8-yard line, and five penalties in a row are given that move the ball half the distance to the (closest) goal. Write a sequence of five terms that specify the location of the ball after each penalty.

Infestation. Leona planted 9 acres of soybeans, but by the end of each week, insects had destroyed one-third of the acreage that was healthy at the beginning of the week. How many acres does she have left after 6 weeks?

The MSRP for a well-equipped 2007 Ford \(\mathrm{F}-250\) Lariat \(4 \mathrm{WD}\) Super Duty Super Cab was \(\$ 43,568\) (www.edmunds.com). Suppose that the price of this model increases by \(5 \%\) each year. Find the price to the nearest dollar for the 2008 through 2012 models.

Discussion. Find the trinomial expansion for \((a+b+c)^{3}\) by using \(x=a\) and \(y=b+c\) in the binomial theorem.

To assess the economic impact of a factory on a community, economists consider the annual amount the factory spends in the community, then the portion of the money that is respent in the community, then the portion of the respent money that is respent in the community, and so on. Suppose a garment manufacturer spends \(\$ 1\) million annually in its community and \(80 \%\) of all money received in the community is respent in the community. Find the first four terms of the economic impact sequence.

Discussion. Find the fourth term in the binomial expansion for \((x+y)^{120} .\) Find the fifth term in the binomial expansion for \((x-2 y)^{100} .\) Did you have any trouble computing the coefficients?

A fabric designer must take into account the capability of textile machines to produce material with vertical repeats. A textile machine can be set up for a vertical repeat every \(\frac{27}{n}\) inches (in.), where \(n\) is a natural number. Write the first five terms of the sequence \(a_{n}=\frac{27}{n},\) which gives the possible vertical repeats for a textile machine.

The note middle C on a piano is tuned so that the string vibrates at 262 cycles per second, or 262 hertz (Hz). The C note one octave higher is tuned to \(524 \mathrm{Hz}\). The tuning for the 11 notes in between using the method called equal temperament is determined by the sequence \(a_{n}=262 \cdot 2^{n / 12} .\) Find the tuning for the 11 notes in between. Round to the nearest whole Hz.

Everyone has two (biological) parents, four grandparents, eight great- grandparents, 16 great-great-grandparents, and so on. If we put the word "great" in front of the word "grandparents" 35 times, then how many of this type of relative do you have? Is this more or less than the present population of the earth? Give reasons for your answers.

If you deposit 1 cent into your piggy bank on September 1 and each day thereafter deposit twice as much as on the previous day, then how much will you be depositing on September \(30 ?\) The total amount deposited for the month can be found without adding up all 30 deposits. Look at how the amount on deposit is increasing each day and see whether you can find the total for the month. Give reasons for your answers.

Use a series to model the situation in each of the following problems. A frog with a vision problem is 1 yard away from a dead cricket. He spots the cricket and jumps halfway to the cricket. After the frog realizes that he has not reached the cricket, he again jumps halfway to the cricket. Write a series in summation notation to describe how far the frog has moved after nine such jumps.

Working in groups, have someone in each group make up a formula for \(a_{n},\) the \(n\)th term of a sequence, but do not show it to the other group members. Write the terms of the sequence on a piece of paper one at a time. After each term is given, ask whether anyone knows the next term. When the group can correctly give the next term, ask for a formula for the \(n\)th term.

Find the sum of each infinite geometric series. $$ \sum_{i=1}^{\infty} 7(0.4)^{i-1} $$

Consider the sequence whose \(n\) th term is \(a_{n}=(0.999)^{n}\). a) Calculate \(a_{100}, a_{1000},\) and \(a_{10,000}\). b) What happens to \(a_{n}\) as \(n\) gets larger and larger?

Use a series to model the situation in each of the following problems. Suppose you earn \(\$ 1\) on January \(1 . \$ 2\) on January \(2, \$ 3\) on January \(3,\) and so on. Use summation notation to write the sum of your earnings for the entire month of January.

The first two terms of the Fibonacci sequence are 0 and \(1 .\) Every term thereafter is the sum of the two previous terms. So the third term is \(1,\) the fourth term is 2 the fifth term is \(3,\) and the sixth term is \(5 .\) So the first 6 terms of the Fibonacci sequence are \(0,1,1,2,3,5\). a) Write the first 10 terms of the Fibonacci sequence. b) Find an application of the Fibonacci sequence by doing a search on the Internet.

What is the difference between a sequence and a series?

Solve each problem using the ideas of arithmetic sequences and series. Increasing salary. If a lab technician has a salary of \(\$ 22,000\) her first year and is due to get a \(\$ 500\) raise each year, then what will her salary be in her seventh year? (IMAGE CANT COPY)

Solve each problem using the ideas of arithmetic sequences and series. On the first day of October an English teacher suggests to his students that they read five pages of a novel and every day thereafter increase their daily reading by two pages. If his students follow this suggestion, then how many pages will they read during October?

Solve each problem using the ideas of arithmetic sequences and series. If an air-conditioning system is not completed by the agreed upon date, the contractor pays a penalty of \(\$ 500\) for the first day that it is overdue, \(\$ 600\) for the second day, \(\$ 700\) for the third day, and so on. If the system is completed 10 days late, then what is the total amount of the penalties that the contractor must pay?

Which of the following sequences is not an arithmetic sequence? Explain your answer. a) \(\frac{1}{2}, 1, \frac{3}{2}, \dots\) b) \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\) c) \(5,0,-5, \dots\) d) \(2,3,4, \ldots\)

What is the smallest value of \(n\) for which \(\sum_{i=1}^{n} \frac{i}{2}>50 ?\)

Which of the following sequences is not a geometric sequence? Explain your answer. a) \(1,2,4, \ldots\) b) \(0.1,0.01,0.001, \ldots\) c) \(-1,2,-4, \dots\) d) \(2,4,6, \dots\)

Write the repeating decimal number \(0.24242424 \ldots\) as an infinite geometric series. Find the sum of the geometric series.