Chapter 12

Use a formula to find the sum of each series. $$\sum_{k=1}^{4}-2\left(\frac{1}{2}\right)^{k}$$

Find the sum for each series. $$\sum_{i=2}^{3} 2(3)^{i}$$

A die is rolled 12 times. Approximate the probability of rolling the following. Exactly 6 ones

Find the sum of the first 10 terms of each arithmetic sequence. $$8,6,4, \dots$$

Use the fundamental principle of counting or permutations to solve each problem. How many 7-digit telephone numbers are possible if the first digit cannot be 0 and (a) only odd digits may be used? (b) the telephone number must be a multiple of 10 (that is, it must end in 0 )? (c) the telephone number must be a multiple of \(100 ?\) (d) the first 3 digits are \(481 ?\) (e) no repetitions are allowed?

Write the indicated tem of each binomial expansion. Twelfth term of \(\left(2 x+y^{2}\right)^{16}\)

Use a formula to find the sum of each series. $$\sum_{j=1}^{3}-3\left(\frac{1}{4}\right)^{j}$$

Find the sum for each series. $$\sum_{i=-1}^{2} 5(2)^{i}$$

A die is rolled 12 times. Approximate the probability of rolling the following. No more than 3 ones

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=10, a_{10}=5.5$$

Use the fundamental principle of counting or permutations to solve each problem. In an experiment on social interaction, 6 people will sit in 6 seats in a row. In how many ways can this be done?

Write the indicated tem of each binomial expansion. Fifteenth term of \(\left(x-y^{3}\right)^{20}\)

CONCEPT CHECK Under what conditions does the sum of the terms of an infinite geometric sequence exist?

Find the sum for each series. $$\sum_{i=-1}^{5}\left(i^{2}-2 i\right)$$

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=-8, a_{10}=-1.25$$

Use the fundamental principle of counting or permutations to solve each problem. In how many ways can 7 of 10 chemicals be added to a beaker for an experiment? (Assume that the order in which the chemicals are add is important.)

Write the indicated tem of each binomial expansion. Tenth term of \(\left(a^{3}+3 b\right)^{11}\)

CONCEPT CHECK The number 0.999 ... can be written as the sum of the terms of an infinite geometric sequence: \(0.9+0.09+0.009+\cdots\) Here we have \(a_{1}=0.9\) and \(r=0.1 .\) Use the formula for \(S_{\infty}\) to find this sum.

Find the sum for each series. $$\sum_{i=3}^{6}\left(2 i^{2}+1\right)$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{20}=1090, a_{20}=102$$

Use the fundamental principle of counting or permutations to solve each problem. A business school offers courses in keyboarding, spreadsheets, transcription, business English, technical writing, and accounting. In how many ways can a student arrange a schedule if 3 courses are taken?

Work each of the following. Find the middle term of \(\left(3 x^{7}+2 y^{3}\right)^{8}\)

Find the sum for each series. $$\sum_{i=1}^{5}\left(3^{i}-4\right)$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{31}=5580, a_{31}=360$$

Use the fundamental principle of counting or permutations to solve each problem. If your college offers 400 courses, 20 of which are in mathematics, and your counselor arranges your schedule of 4 courses by random selection, how many schedules are possible that do not include a math course?

Work each of the following. Find the two middle terms of \(\left(-2 m^{-1}+3 n^{-2}\right)^{11}\)

Write the sum of each geometric series as a rational number. (See Exercise 44.) $$0.7+0.07+0.007+0.0007+\cdots$$

Find the sum for each series. $$\sum_{i=1}^{4}\left[(-2)^{i}-3\right]$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{12}=-108, a_{12}=-19$$

Use the fundamental principle of counting or permutations to solve each problem. In a club with 15 members, how many ways can a slate of 3 officers consisting of president, vice-president, and secretary/treasurer be chosen?

Find the value of \(n\) for which the coefficients of the fifth and eighth terms in the expansion of \((x+y)^{n}\) are the same.

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{5} x_{i}$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{25}=650, a_{25}=62$$

Use the fundamental principle of counting or permutations to solve each problem. A baseball team has 20 players. How many 9 -player batting orders are possible?

Work each of the following. Find the term in the expansion of \((3+\sqrt{x})^{11}\) that con\(\operatorname{tains} x^{4}\)

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{5}-x_{i}$$

Color-Blind Males The probability that a male will be color blind is \(0.042 .\) Approximate the probabilities that in a group of 53 men, the following are true. A. Exactly 5 are color blind. B. No more than 5 are color blind. C. At least 1 is color blind.

Use the fundamental principle of counting or permutations to solve each problem. In how many ways can 5 players be assigned to the 5 positions on a basketball team, assuming that any player can play any position? In how many ways can 10 players be assigned to the 5 positions?

The factorial of a positive integer \(n\) can be computed as a product: \(n !=1 \cdot 2 \cdot 3 \cdot \cdots \cdot n\) Calculators and computers can evaluate factorials quickly. Before the days of techmology. mathematicians developed a formula, called Stirling's formula, for approximating large factorials. Interestingly enough, it imolves the irrational numbers \(\pi\) and \(e\). $$n ! \approx \sqrt{2 \pi n} \cdot n^{n} \cdot e^{-n}$$ As an example, the exact value of \(5 !\) is \(120,\) and Stirling's formula gives the approximation as 118.019168 with a graphing calculator. This is " off" by less than \(2,\) an error of only \(1.65 \%\) Use a calculator and Stirling's formula to find the exact value of \(10 !\) and its approximation.

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{5}\left(2 x_{i}+3\right)$$

Use the fundamental principle of counting or permutations to solve each problem. How many ways can all the letters of the word ELTON be arranged?

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{4}\left(4-6 x_{i}\right)$$

MODELING Disease Infection What will happen when an infectious disease is introduced into a family? Suppose a family has \(I\) infected members and \(S\) members who are not infected, but are susceptible to contracting the disease. The probability \(P\) of \(k\) people not contracting the disease during a l-week period can be calculated by the formula $$P=\left(\begin{array}{l} S \\ k \end{array}\right) q^{k}(1-q)^{s-k}$$ where \(q=(1-p)^{l}\) and \(p\) is the probability that a susceptible person contracts the disease from an infected person. For example, if \(p=0.5,\) then there is a \(50 \%\) chance that a susceptible person exposed to one infected person for 1 week will contract the disease. (Source: Hoppensteadt, F. and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag.) A. Approximate the probability \(P\) of 3 family members not becoming infected within 1 week if there are currently 2 infected and 4 susceptible members. Assume that \(p=0.1\) B. A highly infectious disease can have \(p=0.5 .\) Repeat part (a) with this value of \(p\) C. Approximate the probability that everyone would become sick in a large family if initially \(I=1, S=9\) and \(p=0.5\)

Solve each problem involving combinations. \(A\) banker's association has 30 members. If 4 members are selected at random to present a seminar, how many different groups of 4 are possible?

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$-48,-24,-12,-6, \dots$$

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{3}\left(3 x_{i}-x_{i}^{2}\right)$$

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$625,125,25,5, \dots$$

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{3}\left(x_{i}^{2}+1\right)$$

Solve each problem involving combinations. Howard's Hamburger Heaven sells hamburgers with cheese, relish, lettuce, tomato, mustard, or ketchup. How many different hamburgers can be made that use any 3 of the extras?

It can be shown that $$(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\cdots$$ for any real number \(n\) (not just positive integer values) and any real number \(x\), where \(|x|<1\). Use this result to approximate each quantity to the nearest thousandth. $$(1.02)^{-3}$$