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CHAPTER 19 REVIEW EXERCISES

CONCEPT CHECK EXERCISES

Determine each of the following as being either true or false. If it is false, explain why.

For an arithmetic sequence, if $a_{1} = 5, d = 3,$ $a_{1} = 5, d = 3,$ and $n = 10,$ $n = 10,$ then $a_{10} = 35.$ $a_{10} = 35.$
For a geometric sequence, if $a_{3} = 8, a_{5} = 2,$ $a_{3} = 8, a_{5} = 2,$ then $r = \frac{1}{2} .$ $r = \frac{1}{2} .$
The sum of the geometric series $6 - 2 + \frac{4}{3} - \dots is \frac{9}{2} .$ $6 - 2 + \frac{4}{3} - \dots is \frac{9}{2} .$
Evaluating the first three terms, ${(x + y)}^{10} = x^{10} + 10 x^{9} y + 45 x^{8} y^{2} + \dots$ ${(x + y)}^{10} = x^{10} + 10 x^{9} y + 45 x^{8} y^{2} + \dots$

PRACTICE AND APPLICATIONS

In Exercises 5–12, find the indicated term of each sequence.

1, 6, 11, … (17th)
$1, - 3, - 7, \dots$ $1, - 3, - 7, \dots$ (21st)
500, 100, 20, … (9th)
0.025, 0.01, 0.004, … (7th)
$- 1, 3.5, 8, \dots$ $- 1, 3.5, 8, \dots$ (16th)
$- 1, - \frac{5}{3}, - \frac{7}{3}, \dots$ $- 1, - \frac{5}{3}, - \frac{7}{3}, \dots$ (25th)
$\frac{3}{4}, \frac{1}{2}, \frac{1}{3}, \dots$ $\frac{3}{4}, \frac{1}{2}, \frac{1}{3}, \dots$ (7th)
$5^{- 2}, 5^{0}, 5^{2}, \dots$ $5^{- 2}, 5^{0}, 5^{2}, \dots$ (18th)

In Exercises 13–16, find the sum of each sequence with the indicated values.

$a_{1} = - 4, n = 15, a_{15} = 17$ $a_{1} = - 4, n = 15, a_{15} = 17$ (arith.)
$a_{1} = 300, d = - \frac{20}{3}, n = 10$ $a_{1} = 300, d = - \frac{20}{3}, n = 10$
$a_{1} = 16, r = - \frac{1}{2}, n = 14$ $a_{1} = 16, r = - \frac{1}{2}, n = 14$
$a_{1} = 64, a_{n} = 729, n = 7$ $a_{1} = 64, a_{n} = 729, n = 7$ (geom., $r > 0$ $r > 0$ )

In Exercises 17–26, find the indicated quantities for the appropriate sequences.

$a_{1} = 17, d = - 2, n = 9, S_{9} = ?$ $a_{1} = 17, d = - 2, n = 9, S_{9} = ?$
$d = \frac{4}{3}, a_{1} = - 3, a_{n} = 17, n = ?$ $d = \frac{4}{3}, a_{1} = - 3, a_{n} = 17, n = ?$
$a_{1} = 4, r = \sqrt{2}, n = 7, a_{7} = ?$ $a_{1} = 4, r = \sqrt{2}, n = 7, a_{7} = ?$
$a_{n} = \frac{49}{8}, r = - \frac{2}{7}, S_{n} = \frac{1911}{32}, a_{1} = ?$ $a_{n} = \frac{49}{8}, r = - \frac{2}{7}, S_{n} = \frac{1911}{32}, a_{1} = ?$
$a_{1} = 80, a_{n} = - 25, S_{n} = 220, d = ?$ $a_{1} = 80, a_{n} = - 25, S_{n} = 220, d = ?$
$a_{6} = 3, d = 0.2, n = 11, S_{11} = ?$ $a_{6} = 3, d = 0.2, n = 11, S_{11} = ?$
$n = 6, r = - 0.25, S_{6} = 204.75, a_{6} = ?$ $n = 6, r = - 0.25, S_{6} = 204.75, a_{6} = ?$
$a_{1} = 10, r = 0.1, S_{n} = 11.111, n = ?$ $a_{1} = 10, r = 0.1, S_{n} = 11.111, n = ?$
$a_{1} = - 1, a_{n} = 32, n = 12, S_{12} = ?$ $a_{1} = - 1, a_{n} = 32, n = 12, S_{12} = ?$ (arith.)
$a_{2} = 800, a_{n} = 6400, S_{n} = 32, 500, d = 700, n = ?$ $a_{2} = 800, a_{n} = 6400, S_{n} = 32, 500, d = 700, n = ?$ (arith.)

In Exercises 27–30, find the sums of the given infinite geometric series.

$0.9 + 0.6 + 0.4 + \dots$ $0.9 + 0.6 + 0.4 + \dots$
$1280 - 320 + 80 - \dots$ $1280 - 320 + 80 - \dots$
$1 + {1.02}^{- 1} + {1.02}^{- 2} + \dots$ $1 + {1.02}^{- 1} + {1.02}^{- 2} + \dots$
$3 - \sqrt{3} + 1 - \dots$ $3 - \sqrt{3} + 1 - \dots$

In Exercises 31–34, find the fractions equal to the given decimals.

0.030303 …
0.363363 …
0.0727272 …
0.25399399399 …

In Exercises 35–38, expand and simplify the given expression. In Exercises 39–42, find the first four terms of the appropriate expansion.

${(x - 5)}^{4}$ ${(x - 5)}^{4}$
${(3 + 0.1)}^{4}$ ${(3 + 0.1)}^{4}$
${(x^{2} + 4)}^{5}$ ${(x^{2} + 4)}^{5}$
${(3 n^{1/2} - 2 a)}^{6}$ ${(3 n^{1/2} - 2 a)}^{6}$
${(a + 3 e)}^{10}$ ${(a + 3 e)}^{10}$
${(\frac{x}{4} - y)}^{12}$ ${(\frac{x}{4} - y)}^{12}$
${(p^{2} - \frac{q}{6})}^{9}$ ${(p^{2} - \frac{q}{6})}^{9}$
${(2 s^{2} - \frac{3}{2} t^{- 1})}^{14}$ ${(2 s^{2} - \frac{3}{2} t^{- 1})}^{14}$

In Exercises 43–50, find the first four terms of the indicated expansions by use of the binomial series.

${(1 + x)}^{12}$ ${(1 + x)}^{12}$
${(1 - 2 x)}^{10}$ ${(1 - 2 x)}^{10}$
$\sqrt{1 + x^{2}}$ $\sqrt{1 + x^{2}}$
${(4 - 4 \sqrt{x})}^{- 1}$ ${(4 - 4 \sqrt{x})}^{- 1}$
$\sqrt{1 - a^{2}}$ $\sqrt{1 - a^{2}}$
$\sqrt{1 + 2 b^{4}}$ $\sqrt{1 + 2 b^{4}}$
${(2 - 4 x)}^{- 3}$ ${(2 - 4 x)}^{- 3}$
${(1 + 4 x)}^{- 1/4}$ ${(1 + 4 x)}^{- 1/4}$

In Exercises 51–98, solve the given problems by use of an appropriate sequence or series. All numbers are accurate to at least two significant digits.

Find the sum of the first 1000 positive even integers.
Find the sum of the first 300 positive odd integers.
What is the fifth term of the arithmetic sequence in which the first term is a and the second term is b?
Find three consecutive numbers in an arithmetic sequence such that their sum is 15 and the sum of their squares is 77.
For a geometric sequence, is it possible that $a_{3} = 6, a_{5} = 9,$ $a_{3} = 6, a_{5} = 9,$ and $a_{7} = 12 ?$ $a_{7} = 12 ?$
Find three consecutive numbers in a geometric sequence such that their product is 64 and the sum of their squares is 84.
Find a formula with variable n of the arithmetic sequence with $a_{1} = - 5, a_{n + 1} = a_{n} - 3,$ $a_{1} = - 5, a_{n + 1} = a_{n} - 3,$ for $n = 1, 2, 3, \dots .$ $n = 1, 2, 3, \dots .$
Find the ninth term of the sequence $(a + 2 b), b, - a, \dots .$ $(a + 2 b), b, - a, \dots .$
Approximate the value of ${(1.06)}^{- 6}$ ${(1.06)}^{- 6}$ by using three terms of the appropriate binomial series. Check using a calculator.
Approximate the value of $\sqrt[4]{0.94}$ $\sqrt[4]{0.94}$ by using three terms of the appropriate binomial series. Check using a calculator.
Approximate the value of $\sqrt{30}$ $\sqrt{30}$ by noting that $\sqrt{30} = \sqrt{25 (1.2)} = 5 \sqrt{1 + 0.2}$ $\sqrt{30} = \sqrt{25 (1.2)} = 5 \sqrt{1 + 0.2}$ and using three terms of the appropriate binomial series.
Approximate $\sqrt[3]{29.7}$ $\sqrt[3]{29.7}$ by using the method of Exercise 61.
Without expanding, evaluate

$\begin{array}{l} {(3 x - 2)}^{4} + 4 {(3 x - 2)}^{3} (5 - 3 x) + 6 {(3 x - 2)}^{2} {(5 - 3 x)}^{2} \\ + 4 (3 x - 2) {(5 - 3 x)}^{3} + {(5 - 3 x)}^{4} \end{array}$ $\begin{array}{l} {(3 x - 2)}^{4} + 4 {(3 x - 2)}^{3} (5 - 3 x) + 6 {(3 x - 2)}^{2} {(5 - 3 x)}^{2} \\ + 4 (3 x - 2) {(5 - 3 x)}^{3} + {(5 - 3 x)}^{4} \end{array}$
If $x, x + 2, x + 3$ $x, x + 2, x + 3$ forms a geometric sequence, find x.
If $x + 1, x^{2} - 2, 4 x - 2$ $x + 1, x^{2} - 2, 4 x - 2$ forms an arithmetic sequence, find x.
If $x + x^{2} + x^{3} + \dots = 2,$ $x + x^{2} + x^{3} + \dots = 2,$ find x.
Each stroke of a pile driver moves a post 2 in. less than the previous stroke. If the first stroke moves the post 24 in., which stroke moves the post 4 in.?
During each hour, an exhaust fan removes 15.0% of the carbon dioxide present in the air in a room at the beginning of the hour. What percent of the carbon dioxide remains after 10.0 h?
Each 1.0 mm of a filter through which light passes reduces the intensity of the light by 12%. How thick should the filter be to reduce the intensity of the light to 20%?
A pile of dirt and ten holes are in a straight line. It is 20 ft from the dirt pile to the nearest hole, and the holes are 8 ft apart. If a backhoe takes two trips to fill each hole, how far must it travel in filling all the holes if it starts and ends at the dirt pile?
A roof support with equally spaced vertical pieces is shown in Figure 19.13. Find the total length of the vertical pieces if the shortest one is 10.0 in. long.

Fig. 19.13
During each microsecond, the current in an electric circuit decreases by 9.3%. If the initial current is 2.45 mA, how long does it take to reach 0.50 mA?
A machine that costs $8600 depreciates 1.0% in value each month. What is its value 5 years after it was purchased?
The level of chemical pollution in a lake is 4.50 ppb (parts per billion). If the level increases by 0.20 ppb in the following month and by 5.0% less each month thereafter, what will be the maximum level?
A piece of paper 0.0040 in. thick is cut in half. These two pieces are then placed one on the other and cut in half. If this is repeated such that the paper is cut in half 40 times, how high will the pile be?
After the power is turned off, an object on a nearly frictionless surface slows down such that it travels 99.9% as far during 1 s as during the previous second. If it travels 100 cm during the first second after the power is turned off, how far does it travel while sliding?
Under gravity, an object falls 16 ft during the first second, 48 ft during the second second, 80 ft during the third second, and so on. How far will it fall during the 20th second?
For the object in Exercise 77, what is the total distance fallen during the first 20 s?
An object suspended on a spring is oscillating up and down. If the first oscillation is 10.0 cm and each oscillation thereafter is 9/10 of the preceding one, find the total distance the object travels if it bounces indefinitely.
During each oscillation, a pendulum swings through 85% of the distance of the previous oscillation. If the pendulum swings through 80.8 cm in the first oscillation, through what total distance does it move in 12 oscillations?
A person invests $1000 each year at the beginning of the year. What is the total value of these investments after 20 years if they earn 7.5% annual interest, compounded semiannually?
A well driller charges $10.00 for drilling the first meter of a well and for each meter thereafter charges 0.20% more than for the preceding meter. How much is charged for drilling a 150-m well?
When new, an article cost $250. It was then sold at successive yard sales at 40% of the previous price. What was the price at the fourth yard sale?
Each side of an equilateral triangle is 2 cm in length. The midpoints are joined to form a second equilateral triangle. The midpoints of the second triangle are joined to form a third equilateral triangle. Find the sum of all of the perimeters of the triangles if this process is continued indefinitely. See Fig. 19.14.

Fig. 19.14
In testing a type of insulation, the temperature in a room was made to fall to $\frac{2}{3}$ $\frac{2}{3}$ of the initial temperature after 1.0 h, to $\frac{2}{5}$ $\frac{2}{5}$ of the initial temperature after 2.0 h, to $\frac{2}{7}$ $\frac{2}{7}$ of the initial temperature after 3.0 h, and so on. If the initial temperature was $50.0 ° C,$ $50.0 ° C,$ what was the temperature after 12.0 h? (This is an illustration of a harmonic sequence.)
Two competing businesses make the same item and sell it initially for $100. One increases the price by $8 each year for 5 years, and the other increases the price by 8% each year for 5 years. What is the difference in price after 5 years?
In hydrodynamics, while studying compressible fluid flow, the expression ${(1 + \frac{a - 1}{2} m^{2})}^{a / (a - 1)}$ ${(1 + \frac{a - 1}{2} m^{2})}^{a / (a - 1)}$ arises. Find the first three terms of the expansion of this expression.
In finding the partial pressure $P_{F}$ $P_{F}$ of the fluorine gas under certain conditions, the equation

$P_{F} = \frac{(1 + 2 \times 10^{- 10}) - \sqrt{1 + 4 \times 10^{- 10}}}{2} atm$ $P_{F} = \frac{(1 + 2 \times 10^{- 10}) - \sqrt{1 + 4 \times 10^{- 10}}}{2} atm$

is found. By using three terms of the expansion for $\sqrt{1 + x},$ $\sqrt{1 + x},$ approximate the value of this expression.
During 1 year a beach eroded 1.2 m to a line 48.3 m from the wall of a building. If the erosion is 0.1 m more each year than the previous year, when will the waterline reach the wall?
On a highway with a steep incline, the runaway truck ramp is constructed so that a vehicle that has lost its brakes can stop. The ramp is designed to slow a truck in succeeding 20-m distances by 10 km/h, 12 km/h, 14 km/h, …. If the ramp is 160 m long, will it stop a truck moving at 120 km/h when it reaches the ramp?
Each application of an insecticide destroys 75% of a certain insect. How many applications are needed to destroy at least 99.9% of the insects?
At the end of 2005 a house was valued at $375,000, and then increased in value by 5.00%, on average, each year 2006 through 2011. Its value did not change in 2012, but then decreased by 8.00%, on average, each year 2013 through 2016. What was its value at the end of 2016?
Oil pumped from a certain oil field decreases 10% each year. How long will it take for production to be 10% of the first year’s production?
A wire hung between two poles is parabolic in shape. To find the length of wire between two points on the wire, the expression $\sqrt{1 + 0.08 x^{2}}$ $\sqrt{1 + 0.08 x^{2}}$ is used. Find the first three terms of the binomial expansion of this expression.
Show that the middle term of a finite arithmetic sequence equals S/n, if n is odd.
The sum of three terms of a geometric sequence is $- 3,$ $- 3,$ and the second is 6 more than the first. Find these terms.
Do the reciprocals of the terms of a geometric sequence form a geometric sequence? Explain.
The terms $a, a + 12, a + 24$ $a, a + 12, a + 24$ form an arithmetic sequence, and the terms $a, a + 24, a + 12$ $a, a + 24, a + 12$ form a geometric sequence. Find these sequences.
Derive a formula for the value V after 1 year of an amount A invested at r% (as a decimal) annual interest, compounded n times during the year. If $A = $1000$ $A = $1000$ and $r = 0.10$ $r = 0.10$ (10%), write two or three paragraphs explaining why the amount of interest increases as n increases and stating your approach to finding the maximum possible amount of interest.

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Table of Contents for CHAPTER 19 REVIEW EXERCISES

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Table of Contents for
CHAPTER 19 REVIEW EXERCISES