Chapter 8

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

Use a formula to find the sum of each arithmetic series. $$3+5+7+9+11+13+15+17$$

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 1-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 infectious person. For example, if \(p=0.5,\) then there is a \(50 \%\) chance that a susceptible person exposed to one infectious 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 \(p=0.5\)

Find each sum that converges. $$16+2+\frac{1}{4}+\frac{1}{32}+\cdots$$

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

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

Use a formula to find the sum of each arithmetic series. $$7.5+6+4.5+3+1.5+0+(-1.5)$$

Find each sum that converges. $$18+6+2+\frac{2}{3}+\dots$$

\text {Solve each problem involving combinations.} Apple Samples How many different samples of 3 apples can be drawn from a crate of 25 apples?

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

Use a formula to find the sum of each arithmetic series. $$1+2+3+4+\dots+50$$

Find each sum that converges. $$100+10+1+\cdots$$

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 in Exercises \(53-56\) to the nearest thousandth. $$(1.02)^{-3}$$

\text {Solve each problem involving combinations.} Hamburger Choices 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?

Evaluate the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for each function. $$f(x)=4 x-7$$

Use a formula to find the sum of each arithmetic series. $$1+3+5+7+\cdots+97$$

Find each sum that converges. $$128+64+32+\cdots$$

\text {Solve each problem involving combinations.} Financial Planners Three financial planners are to be selected from a group of 12 to participate in a special program. In how many ways can this be done? In how many ways can the group that will not participate be selected?

Evaluate the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for each function. $$f(x)=6+2 x$$

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 in Exercises \(53-56\) to the nearest thousandth. $$\frac{1}{1.04^{5}}$$

Use a formula to find the sum of each arithmetic series. $$-7+(-4)+(-1)+2+5+\cdots+98+101$$

\text {Solve each problem involving combinations.} Marble Samples If a bag contains 15 marbles, how many samples of 2 marbles can be drawn from it? how many samples of 4 marbles?

Evaluate the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for each function. $$f(x)=2 x^{2}$$

Find each sum that converges. $$\frac{4}{3}+\frac{2}{3}+\frac{1}{3}+\dots$$

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 in Exercises \(53-56\) to the nearest thousandth. $$(1.01)^{3 / 2}$$

Use a formula to find the sum of each arithmetic series. $$89+84+79+74+\cdots+9+4$$

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 in Exercises \(53-56\) to the nearest thousandth. $$(1.03)^{0.2}$$

\text {Solve each problem involving combinations.} Card Combinations Five cards marked respectively with the numbers \(1,2,3,4,\) and 5 are shuffled, and 2 cards are then drawn. How many different 2-card hands are possible?

Evaluate the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for each function. $$f(x)=x^{2}-1$$

Use a formula to find the sum of each arithmetic series. The first 40 terms of the series \(a_{n}=5 n\)

Find each sum that converges. $$\sum_{i=1}^{\infty} 3\left(\frac{1}{4}\right)^{i-1}$$

\text {Solve each problem involving combinations.} Marble Samples \(\quad\) In Exercise \(55,\) if the bag contains 3 yellow, 4 white, and 8 blue marbles, how many samples of 2 can be drawn in which both marbles are blue?

Evaluate the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for each function. $$f(x)=\frac{-2}{x+1}$$

Use a formula to find the sum of each arithmetic series. The first 50 terms of the series \(a_{n}=1-3 n\)

Find each sum that converges. $$\sum_{i=1}^{\infty} 5\left(-\frac{1}{4}\right)^{i-1}$$

Evaluate the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for each function. $$f(x)=\frac{5}{2 x-1}$$

Evaluate each sum. $$\sum_{i=1}^{3}(i+4)$$

Find each sum that converges. $$\sum_{k=1}^{\infty}(0.3)^{k}$$

\text {Solve each problem involving combinations.} Convention Delegation Choices A city council is composed of 5 liberals and 4 conservatives. Three members are to be selected randomly as delegates to a convention. (a) How many delegations are possible? (b) How many delegations could have all liberals? (c) How many delegations could have 2 liberals and 1 conservative? (d) If 1 member of the council serves as mayor, how many delegations are possible that include the mayor?

Find the sum for each series. $$\sum_{i=1}^{100} 6$$

Evaluate each sum. $$\sum_{i=1}^{5}(i-8)$$

Find each sum that converges. $$\sum_{k=1}^{\infty}(0.1)^{k}$$

\text {Solve each problem involving combinations.} Delegation Choices Seven workers decide to send a delegation of 2 to their supervisor to discuss their grievances. (a) How many different delegations are possible? (b) If it is decided that a certain employee must be in the delegation, how many different delegations are possible? (c) If there are 2 women and 5 men in the group, how many delegations would include at least 1 woman?

Find the sum for each series. $$\sum_{i=1}^{20} \frac{1}{2}$$

Evaluate each sum. $$\sum_{j=1}^{10}(2 j+3)$$

Find each sum that converges. $$\sum_{k=1}^{\infty} 5^{-k}$$

Use any or all of the methods described in this section to solve each problem. Course Schedule If Kim Falgout has 8 courses to choose from, how many ways can she arrange her schedule if she must pick 4 of them?

Find the sum for each series. $$\sum_{i=1}^{15} i^{2}$$

Evaluate each sum. $$\sum_{j=1}^{15}(5 j-9)$$

Find each sum that converges. $$\sum_{k=1}^{\infty} 3^{-k}$$