Exam 11: Sequences, Induction, and Probability

Use the Formula for the General Term of a Geometric Sequence -Find $\mathrm { a } _ { 12 }$ when $\mathrm { a } _ { 1 } = 4000 , \mathrm { r } = \frac { 1 } { 3 }$ .

Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove that the statement is true for every positive integer n. - $2 \text { is a factor of } n ^ { 2 } - n + 2$

Write the first four terms of the sequence whose general term is given. - $a _ { n } = 4 ( 3 n - 1 )$

Use Summation Notation - $\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k } ( k + 4 )$

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. - $S _ { n } : 2 \text { is a factor of } n ^ { 2 } + 11 n$ A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely.

Find the Probability of One Event and a Second Event Occurring -Urn A has balls numbered 1 through 8 . Urn B has balls numbered 1 through 3 . What is the probability that a 4 is drawn from A followed by a 2 from B?

Use the Permutations Formula -How many 2-letter codes can be formed using the letters A, B, C, D, E, and F? No letter can be used more than once.

Use Factorial Notation - $a _ { n } = \frac { 3 ^ { n } } { ( n + 2 ) ! }$

Write the first four terms of the sequence whose general term is given. -A deposit of $\$ 6000$ is made in an account that earns $9 \%$ interest compounded quarterly. The balance in the account after $n$ quarters is given by the sequence $a _ { n } = 6000 \left( 1 + \frac { 0.09 } { 4 } \right) ^ { n } \quad n = 1,2,3 , \ldots$ Find the balance in the account after 7 years.

Use the Combinations Formula -A hamburger shop sells hamburgers with cheese, relish, lettuce, tomato, onion, mustard, or ketchup. How many different hamburgers can be concocted using any 3 of the extras?

Arithmetic Sequences Find the Common Difference for an Arithmetic Sequence - $- 15 , - 17 , - 19 , - 21 , \ldots$

Use the Formula for the General Term of a Geometric Sequence -Find $a _ { 6 }$ when $a _ { 1 } = 6 , r = 4$ .

Distinguish Between Permutation Problems and Combination Problems -In a student government election, 5 seniors, 2 juniors, and 3 sophomores are running for election. Students elect four at-large senators. In how many ways can this be done?

Use the Formula for the General Term of an Arithmetic Sequence Choose the one alternative that best completes the statement or answers the question. Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find $a 18$ when $a _ { 1 } = 10 , d = - 1$ .

Distinguish Between Permutation Problems and Combination Problems -A club elects a president, vice-president, and secretary-treasurer. How many sets of officers are possible if there are 11 members and any member can be elected to each position? No person can hold more than one office.

Geometric Sequences and Series 1 Find the Common Ratio of a Geometric Sequence - $\frac { 3 } { 3 } , \frac { 3 } { 12 } , \frac { 3 } { 48 } , \frac { 3 } { 192 } , \frac { 3 } { 768 }$

Use the Formula for the General Term of a Geometric Sequence -Find $a _ { 5 }$ when $a _ { 1 } = 7 , r = - 3$ .

Find the Probability of One Event or a Second Event Occurring -A card is drawn from a deck of 52 cards. What is the probability that it is a numbered card (2-10) or a club?

Write Terms of an Arithmetic Sequence - $a _ { 1 } = 14 ; d = - 3$

Write Terms of an Arithmetic Sequence - $a _ { 1 } = - 21 ; d = 4$