Exam 8: Sequences, Series, Induction, and Probability

Solve the problem. -A club must elect a president, vice-president, and treasurer from its 13 members. In how many ways can the positions be filled?

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is arithmetic. If so, find the common difference. -4, -16, 64, -256, . . .

Write the sum using summation notation. - $a + a r + a r ^ { 2 } + a r ^ { 3 } + \ldots + a r ^ { 19 }$

Write a nonrecursive formula for the nth term of the arithmetic sequence {an} based on the given information. - $a _ { 1 } = \frac { 1 } { 2 } , d = \frac { 2 } { 3 }$

Solve the problem. -The yearly salary for job A is $57,000 initially with an annual raise of $3,200 every year thereafter. The yearly salary for job B is $53,000 initially with an annual raise of 5%. a. Consider a sequence representing the salary for job A for year n. Is this an arithmetic or geometric Sequence? Find the total earnings for job A over 20 years. Round to the nearest dollar. b.Consider a sequence representing the salary for job B for year n. Is this an arithmetic or geometric Sequence? Find the total earnings for job B over 20 years. Round to the nearest dollar.

Expand the binomial by using the binomial theorem. - $\left( \frac { a } { 3 } - b \right) ^ { 5 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The number of permutations of n elements taken r at a time is denoted by ${ } _ { n } P _ { r }$ and is computed by ${ } _ { n } P _ { r } = \frac { \square } { \square } \text { or } { } _ { n } P _ { r } = n ( n - 1 ) ( n - 2 ) \ldots ( n - r + 1 )$

Solve the problem. -At a hospital specializing in treating heart disease, it was found that 224 out of 4,612 patients undergoing open heart mitral valve surgery died during surgery or within 30 days after surgery. Determine the probability that a patient will not survive the surgery or 30 days after the surgery. This is called the mortality rate. Round to 3 decimal places.

Find the indicated term of the binomial expansion. - $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 14 }$ ; Find the term containing $x ^ { 33 }$ .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A sequence of payments made at equal intervals over a fixed period of time is called an ________.

Find the indicated term. - $a _ { 1 } = - 11 , d = 9 ;$ Find $a _ { 18 }$

Solve the problem. -Consider an experiment where a die is rolled that has 16 sides. The outcomes are the numbers 1 to 16. Determine the probability of the given event. A number between 3 and 8, inclusive, is rolled.

Find the sum. - $\sum _ { i = 1 } ^ { 4 } ( 5 i + 1 )$

Expand the binomial by using the binomial theorem. - $( 5 y + 1 ) ^ { 4 }$

Provide the missing information. -Consider $( a + b ) ^ { n }$ where n is a whole number. The coefficients of the terms in the expansion can be found by using ________ triangle or by using $\left( \begin{array} { l } n \\r\end{array} \right)$

Find $a _ { 1 }$ and and r for a geometric sequence { $\left\{ a _ { n } \right\}$ } from the given information. - $a _ { 2 } = 18$ and $a _ { 7 } = 4,374$

Use the data in the table categorizing cholesterol levels by the ages of the individuals in a study. If one person from the study is chosen at random, find the probability of the given event. -The person has elevated cholesterol. Normal Cholesterol Elevated Cholesterol Total 30 and under 11 5 16 52 28 80 61 or older 24 80 104 Total 87 113 200

Find the indicated term of the binomial expansion. - $\left( 2 r + s ^ { 2 } \right) ^ { 10 }$ ; eighth term

Use mathematical induction to prove the given statement for all positive integers n. -3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

Solve the problem. -Suppose a tournament has 15 participants. In how many different ways can the 15 players be paired to play in the first round of the tournament? Assume that each player can play any other player Without regard to seeding.