Exam 8: Sequences, Series, Induction, and Probability

Provide the missing information. -The expression $a ^ { 3 } + 3 a ^ { 2 } b + 3 a b ^ { 2 } + b ^ { 3 }$ is called the _________ expansion of $( a + b ) ^ { 3 }$

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. - $a _ { n } = \frac { ( - 1 ) ^ { n } } { n ^ { 2 } + 4 }$

Solve the problem. -Doctors in a certain city report 30 confirmed cases of the flu to the health department. If the number of reported cases increases by roughly 35% each week thereafter, find the number of new cases Reported 7 weeks after the initial report. Round to the nearest whole unit.

Solve the problem. -Find the sum of the first 50 integers that are exactly divisible by 3 .

Solve the problem. -A committee of 4 men and 3 women is to be made from a group of 12 men and 8 women. In how many ways can such a committee be formed?

Solve the problem. -An object in free fall is dropped from a tall cliff. It falls $16 \mathrm { ft }$ in the first second, $48 \mathrm { ft }$ in the second second, $80 \mathrm { ft }$ in the third second, and so on. Write a formula for the nth term of an arithmetic sequence that represents the distance $d _ { n }$ (in $\mathrm { ft }$ ) that the object will fall in the $n$ th second.

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 is between 31 and 60, inclusive, or elevated cholesterol. Normal Cholesterol Elevated Cholesterol Total 30 and under 15 3 18 51 30 81 or older 19 82 101 Total 85 115 200

The nth term of a sequence is given. Find the indicated term. - $a _ { n } = \frac { 2 ^ { n } } { ( n + 1 ) ! } ; a _ { 4 }$

Write the first four terms of a geometric sequence, {an}, based on the given information about the sequence. - $a _ { 1 } = - 5 , r = - 4$

Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove the given statement. - $10 n < 4 ^ { n } \text { for positive integers } \mathrm { n } \geq 3 \text {. }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The nth term of a geometric sequence with first term a1 and common ratio r is given by $a _ { n } =$ ________

Find the sum. - $3 + 4.6 + 6.2 + 7.8 + \ldots + 55.8$

Find the indicated term of the binomial expansion. - $\left( \sqrt { 3 } x ^ { 3 } + y ^ { 5 } \right) ^ { 9 } ;$ fourth term

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. - $a _ { n } = e ^ { 2 \ln n }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An________ is a subset of the sample space of an experiment.

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Consider the set of integers from 1 to 22, inclusive. If one number is selected, in how many ways can we obtain a number that is divisible by 7?

Solve the problem. -A slot machine in a casino has four wheels that all spin independently. Each wheel has 11 stops, denoted by 0 through 9, and bar. What is the probability that a given outcome is bar-bar-bar?

Find the indicated term of a geometric sequence from the given information. - $a _ { 1 } = 24$ and $a _ { 2 } = - 18$ . Find the sixth term.

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. - $a ^ { n } = ( - 1 ) ^ { n + 1 } 3 ^ { n }$

Solve the problem. -In a state lottery, a player wins the grand prize by choosing the same group of 4 numbers from 1 through 45 as is chosen by the computer. How many 4-number groups are possible?