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 arithmetic. If so, find the common difference. - $5 , \frac { 25 } { 6 } , \frac { 125 } { 36 } , \frac { 625 } { 216 }$

Find the indicated term of the arithmetic sequence based on the given information. - $a _ { 1 } = 38 , d = 3$ ; Find $a _ { 25 }$ .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If two events A and B are not mutually exclusive, then $P ( A \cup B )$ ) can be computed by the formula $P ( A \cup B ) =$ .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A ________ sequence is a sequence in which each term after the first is the product of the preceding term and a fixed nonzero real number, called the common________ , r.

Expand the binomial by using the binomial theorem. - $\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -A teacher is going to assign each student a 3-digit code using the digits 0 through 4. How many codes are possible if no repetitions are allowed?

Find the sum. - $\sum _ { k = 1 } ^ { 6 } \left( - 4 k ^ { 3 } \right)$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Suppose you wish to prove the statement that follows using the extended principle of mathematical is 2 is a factor of $5 ^ { n } - 3$ , for all natural numbers $n$ . Assume that $S _ { k }$ is true for $\mathrm { k } \geq 1$ where $S _ { k }$ is the statement that 2 is a factor of $5 ^ { k } - 3$ , and write the statement $S _ { k + 1 }$ .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The sum of the first n terms of a sequence is called the nth ________ sum and is denoted by $S _ { n }$

Evaluate. -a. In how many ways can the letters in the word TEXAS be arranged? b. In how many ways can the letters in the word MISSISSIPPI be arranged?

Find the indicated term of the arithmetic sequence based on the given information. - $a _ { 1 } = \frac { 4 } { 5 } , d = \frac { 1 } { 3 } ;$ Find $a _ { 24 }$

Solve the problem. -Suppose you wish to prove the statement that follows using the extended principle of mathematical is $( n + 1 ) ! > 3 ^ { n }$ , for positive integers $n \geq 4$ . Assume that $S _ { k }$ is true for $k \geq 4$ where $S _ { k }$ is the statement $( n + 1 ) ! > 3 ^ { n }$ , and write the statement $S _ { k + 1 }$

A ________formula defines the nth term of a sequence as a function of one or more terms preceding it.

Find the indicated term of the arithmetic sequence based on the given information. - $a _ { 1 } = 13$ and $a _ { 18 } = - 55 ;$ Find $a _ { 9 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The difference between consecutive terms in an arithmetic sequence is called the________ ________, and is denoted by d.

Solve the problem. -Suppose that an individual is paid $0.02 on day 1 and every day thereafter, the payment is doubled. a. Write a formula for the nth term of a sequence that gives the payment (in $) on day n. b. How much will the individual earn on day 10? day 20?

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -A license plate has 2 letters followed by 4 digits. How many license plates can be made if there are no restrictions on the letters or digits.

Use mathematical induction to prove the given statement for all positive integers n. -3 + 5 + 7 + ... + (2n + 1) = n(n + 2)

The nth term of a sequence is given. Find the indicated term. - $a _ { n } = \frac { 3 } { n } - 1 ; a _ { 10 }$

Provide the missing information. - $\text { Given }(a+b)^{17} \text { the } 12 \text { th term is given by }\left(\int a^{\square} b^{\square}\right. \text {. }$