Chapter 12

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$12,24,48,96, \dots$$

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

Solve each problem involving combinations. 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?

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. $$\frac{1}{1.04^{5}}$$

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$2,-10,50,-250, \dots$$

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

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

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

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. $$(1.01)^{3 / 2}$$

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. $$7.5+6+4.5+3+1.5+0+(-1.5)$$

Solve each problem involving 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?

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. $$(1.03)^{0.2}$$

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. $$f(x)=6+2 x$$

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

Solve each problem involving combinations. 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. $$f(x)=2 x^{2}$$

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

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. $$-7+(-4)+(-1)+2+5+\dots+98+101$$

Solve each problem involving combinations. 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?

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. $$89+84+79+74+\cdots+9+4$$

Solve each problem involving combinations. 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?

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. $$f(x)=\frac{5}{2 x-1}$$

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

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

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

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

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

Use any or all of the methods described in this section to solve each problem. How many samples of 3 pineapples can be drawn from a crate of \(12 ?\)

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

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

Find the sum of each series. $$\sum_{i=1}^{3}(i+4)$$

Use any or all of the methods described in this section to solve each problem. A chef specializes in making different vegetable soups with carrots, celery, beans, peas, mushrooms, and potatoes. How many different soups can he make with any 4 ingredients?

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

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

Find the sum of each series. $$\sum_{i=1}^{5}(i-8)$$

Use any or all of the methods described in this section to solve each problem. From a pool of 7 assistants, 3 are selected to be assigned to 3 managers, 1 assistant to each manager. In how many ways can this be done?

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

Find the sum for each series. $$\sum_{i=1}^{50} 2 i^{3}$$

Find the sum of each series. $$\sum_{j=1}^{10}(2 j+3)$$

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

Find the sum of each series. $$\sum_{j=1}^{15}(5 j-9)$$

Use any or all of the methods described in this section to solve each problem. In an experiment on plant hardiness, a researcher gathers 6 wheat plants, 3 barley plants, and 2 rye plants. She wishes to select 4 plants at random. (a) In how many ways can this be done? (b) In how many ways can it be done if exactly 2 wheat plants must be included?

Find the sum for each series. $$\sum_{i=1}^{5}(8 i-1)$$

Find the sum of each series. $$\sum_{i=1}^{12}(-5-8 i)$$

Use any or all of the methods described in this section to solve each problem. In a club with 8 men and 11 women members, how many 5 -member committees can be chosen that have the following? (a) All men (b) All women (c) 3 men and 2 women (d) No more than 3 women

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

Find the sum for each series. $$\sum_{i=1}^{5}\left(4 i^{2}-2 i+6\right)$$