Chapter 9

Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of a polygon.) Octagon

Let \(A\) and \(B\) be two events from the same sample space such that \(P(A)=0.76\) and \(P(B)=0.58\) (a) Is it possible that \(A\) and \(B\) are mutually exclusive? Explain. Draw a diagram to support your answer. (b) Is it possible that \(A^{\prime}\) and \(B^{\prime}\) are mutually exclusive? Explain. Draw a diagram to support your answer. (c) Determine the possible range of \(P(A \cup B)\)

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 10(0.11)^{n}$$

Expand the binomial by using Pascal's Triangle to determine the coefficients. \((5 y+2)^{5}\)

Simplify the factorial expression. $$\frac{(2 n-1) !}{(2 n+1) !}$$

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{250}(1000-n)$$

Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of a polygon.) Decagon (10 sides)

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 5(0.45)^{n}$$

Expand the binomial by using Pascal's Triangle to determine the coefficients. \((2 v+3)^{6}\)

Simplify the factorial expression. $$\frac{(2 n-2) !}{(2 n) !}$$

Use a graphing utility to find the partial sum. $$\sum_{n=1}^{20}(2 n+1)$$

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty}\left[-3(-0.9)^{n}\right]$$

Expand the binomial by using Pascal's Triangle to determine the coefficients. \((2 x+3 y)^{5}\)

Use a graphing utility to find the partial sum. $$\sum_{n=1}^{50}(40-2 n)$$

Solve for \(n\). $$_{n} P_{5}=18 \cdot_{n-2} P_{4}$$

Evaluate \(_{n} C_{r} .\) Verify your result using a graphing utility. $$_{9} C_{5}$$

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty}\left[-10(-0.2)^{n}\right]$$

Expand the binomial by using Pascal's Triangle to determine the coefficients. \((3 x+4 y)^{5}\)

Use a graphing utility to find the partial sum. $$\sum_{n=0}^{100} \frac{n+5}{2}$$

Solve for \(n\). $$_{n} P_{4}=10 \cdot_{n-1} P_{3}$$

Evaluate \(_{n} C_{r} .\) Verify your result using a graphing utility. $$_{11} C_{8}$$

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$9+6+4+\frac{8}{3}+\cdots$$

Expand the binomial by using Pascal's Triangle to determine the coefficients. \((3 t-2 v)^{4}\)

Use a graphing utility to find the partial sum. $$\sum_{n=0}^{100} \frac{4-n}{4}$$

Solve for \(n\). $$_{n} P_{6}=12 \cdot_{n-1} P_{5}$$

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$8+6+\frac{9}{2}+\frac{27}{8}+\cdots$$

Expand the binomial by using Pascal's Triangle to determine the coefficients. \((5 v-2 z)^{4}\)

Solve for \(n\). $$_{n+1} P_{3}=4 \cdot_{n} P_{2}$$

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$3+\frac{15}{2}+\frac{75}{4}+\frac{375}{8}+\cdots$$

Use the Binomial Theorem to expand and simplify the expression. \((3 \sqrt{x}+5)^{3}\)

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=\frac{2}{3} n$$

Use a graphing utility to find the partial sum. $$\sum_{j=1}^{200}(10.5+0.025 j)$$

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$2+\frac{7}{3}+\frac{49}{18}+\frac{343}{108}+\cdots$$

Use the Binomial Theorem to expand and simplify the expression. \((2 \sqrt{t}-7)^{3}\)

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=\frac{1}{2} n+3$$

A brick patio has the approximate shape of a trapezoid, as shown in the figure. The patio has 18 rows of bricks. The first row has 14 bricks and the 18 th row has 31 bricks. How many bricks are in the patio?

Solve for \(n\). $$4 \cdot_{n+1} P_{2}=_{n+2} P_{3}$$

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$-7+2-\frac{4}{7}+\frac{8}{49}-\cdots$$

Use the Binomial Theorem to expand and simplify the expression. \(\left(x^{2 / 3}-y^{1 / 3}\right)^{3}\)

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=16(-0.5)^{n-1}$$

An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on (see figure). How many seats are there in all 20 rows?

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$-6+5-\frac{25}{6}+\frac{125}{36}-\cdots$$

Use the Binomial Theorem to expand and simplify the expression. \(\left(u^{3 / 5}+v^{1 / 5}\right)^{5}\)

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=8(-0.75)^{n-1}$$

A hardware store makes a profit of \(\$ 30,000\) during its first year. The store owner sets a goal of increasing profits by 5000 dollar each year for 4 years. Assuming that this goal is met, find the total profit during the first 5 years of business.

Expand the expression in the difference quotient and simplify. \(\frac{f(x+h)-f(x)}{h}, h \neq 0\) \(f(x)=x^{3}\)

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=\frac{2 n}{n+1}$$

An object with negligible air resistance is dropped from a plane. During the first second of fall, the object falls 16 feet; during the second second, it falls 48 feet; during the third second, it falls 80 feet; and during the fourth second, it falls 112 feet. Assume this pattern continues. How many feet will the object fall in 8 seconds?

Determine whether the statement is true or false. Justify your answer. The number of permutations of \(n\) elements can be derived by using the Fundamental Counting Principle.

Expand the expression in the difference quotient and simplify. \(\frac{f(x+h)-f(x)}{h}, h \neq 0\) \(f(x)=x^{4}\)