Exam 8: Sequences, Series, Induction, and Probability

Use mathematical induction to prove the given statement for all positive integers n. - $1 \cdot 4 + 2 \cdot 7 + 3 \cdot 10 + \ldots + n ( 3 n + 1 ) = n ( n + 1 ) ^ { 2 }$

Write the sum using summation notation. - $\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -A quiz consists of 5 multiple-choice questions, each with five possible responses, and 5 true/false questions. In how many ways can a student answer the questions on the test?

One property of summation indicates that $\sum _ { i = 1 } ^ { n } c =\_\_\_\_.$

Evaluate the expression. - $\frac { 9 ! } { 5 ! \cdot 4 ! }$

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. - $a _ { n } = n \left( n ^ { 3 } - 8 \right)$

Evaluate the expression. - $\left( \begin{array} { l } 8 \\ 3 \end{array} \right)$

Use mathematical induction to prove the given statement for all positive integers n. - $1 + 4 + 4 ^ { 2 } + 4 ^ { 3 } + \ldots + 4 ^ { n - 1 } = \frac { 4 ^ { n } - 1 } { 3 }$

The expression $a _ { n }$ is called the________ term or general term of a sequence.

Choose the one alternative that best completes the statement or answers the question. Use Pascal's triangle to write the expansion of the the given expression. - $( x - y ) ^ { 4 } \text {. }$

Solve the problem. -In how many ways can a manager assign 7 employees at a coffee shop to 7 different tasks? Assume that each employee is assigned to exactly one task.

Find the sum. - $\sum _ { k = 2 } ^ { 5 } \left( \frac { 1 } { 3 } \right) ^ { k }$

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. - $\text { If } 0 < x < 1 \text {, then } x ^ { n } < x ^ { n - 1 } \text {. }$

Write the first five terms of the sequence defined recursively. - $b _ { 1 } = 5 ; b _ { n } = 5 b _ { n - 1 } - 3$

Find the sum. - $\sum _ { n = 2 } ^ { 4 } \left( n ^ { 2 } + 5 \right)$

Write the sum using summation notation. - $- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }$

Find the nth term $a _ { n }$ of a sequence whose first four terms are given. - $\frac { 5 } { 16 } , \frac { 10 } { 25 } , \frac { 15 } { 36 } , \frac { 20 } { 49 } , \ldots$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the formula $S_{n}=$ _______

Find the number of terms of the finite arithmetic sequence. - $12,11.7,11.4,11.1 , \ldots , - 2.4$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Suppose that n represents the number of distinct elements in a group from which r elements will be chosen in a particular order. Then each arrangement is called a________ of n items taken ________ at atime.