Exam 8: Sequences, Series, Induction, and Probability

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is geometric. If so, find the value of r. - $40,5 , \frac { 5 } { 8 } , \frac { 5 } { 64 } , \ldots$

Choose the one alternative that best completes the statement or answers the question. The nth term of a sequence is given. Write the first four terms of the sequence. - $a _ { n } = ( - 1 ) ^ { n } \frac { n + 3 } { n + 4 }$

Solve the problem. -In a "Pick-5" game, a player wins a prize for matching a 5-digit number from 0000 to 99,999 with the number randomly selected during the drawing. How many 5-digit numbers can a player choose? Assume that a number can start with a zero or zeros such as 00001.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An________ sequence is a sequence in which each term after the first differs from its predecessor by a fixed constant.

Solve the problem. -Consider an experiment where a die is rolled that has 15 sides. The outcomes are the numbers 1 to 15. Determine the probability of the given event. A number less than 4 is rolled.

Expand the binomial by using the binomial theorem. - $\left( e ^ { x } - e ^ { - x } \right) ^ { 4 }$

Provide the missing information. -Consider $( a + b ) ^ { n }$ where n is a whole number. How many terms are in the binomial expansion?

Solve the problem. -Consider the sample space when two fair dice are rolled. Determine the probability for the following event. The sum of the numbers on the dice is less than 8.

From the given terms of the arithmetic sequence, find a1 and d. - $a _ { 12 } = 50$ and $a _ { 25 } = 89$

Write the first four terms of an arithmetic sequence, $\left\{ a _ { n } \right\}$ , based on the given information about the sequence. -Write the first five terms of the arithmetic sequence. $a _ { 1 } = 3 , d = - 4$

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 normal cholesterol or is 61 or older. Normal Cholesterol Elevated Cholesterol Total and under 11 5 16 31-60 45 31 76 61 or older 18 90 108 Total 74 126 200

Simplify the difference quotient: $\frac { f ( x + h ) - f ( x ) } { h }$ - $x ^ { 4 } - 3 x ^ { 2 } - 1$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Let $P _ { n }$ be a statement involving the positive integer n, and let k be an arbitrary positive integer. Proof by mathematical indicates that $P _ { n }$ is true for all positive integers n if 1. is true, and 2.The truth of $P _ { k }$ implies the truth of .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A ________diagram can be used to illustrate the possible outcomes in a sequence of events.

Solve the problem. -A farmer depreciates a $100,000 tractor. He estimates that the resale value of the tractor n years after purchase is 83% of its value from the previous year. a. Write a formula for the nth term of a sequence that represents the resale value of the tractor n years After purchase. b. What will the resale value be after 7 years? Round to the nearest $1000.

Find $a _ { 1 }$ and and r for a geometric sequence { $\left\{ a _ { n } \right\}$ } from the given information. - $a _ { 2 } = - 6$ and $a _ { 6 } = - \frac { 243 } { 128 }$

Write a formula for the nth term of the geometric sequence. - $\frac { 64 } { 5 } , 8,5 , \frac { 25 } { 8 }$

Find the indicated term of the arithmetic sequence based on the given information. - $a _ { 1 } = 36$ and $a _ { 6 } = 11 ;$ Find $a _ { 33 }$ .

Provide the missing information. - $\text { For positive integers } n \text { and } k(k \leq n+1) \text {, the } k \text { th term of }(a+b)^{n} \text { is given by }$

Solve the problem. -In a drama class, 5 students are to be selected from 19 students to perform a synchronized dance. In how many ways can 19 students be selected from the 5?