Exam 9: Sequences and Series; Counting and Probability

Use the principle of mathematical induction to show that the mathematical statement is true for all natural numbers n. - $S _ { n } : 5 + \frac { 5 } { 6 } + \frac { 5 } { 36 } + \ldots + \frac { 5 } { 6 ^ { n - 1 } } = 6 \left( 1 - \frac { 1 } { 6 ^ { n } } \right)$

Solve the problem. -Rewrite the number $0 . \overline { 35 }$ as the quotient of two integers.

Write the first four terms of the recursive sequence. - $a _ { 1 } = 2 , a _ { n } = n - a _ { n } - 1$ for $n \geq 2$

Provide an appropriate response. -Find the 8th term of the geometric sequence $1000, $1060, $1123.60, . . . .

Determine if the sequence is arithmetic. If the sequence is arithmetic, find the common difference. - $a _ { n } = \frac { 7 n - 1 } { 8 }$

-A box contains 13 white cards numbered 1 through 13. State the number of elements in the sample space of the event choosing one card with a number greater than 6.

Solve the problem. -An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 5 of the multiple-choice questions and 2 of the essay questions, in how many ways can the questions be chosen?

Find the sum of the series. - $\sum _ { i = 2 } ^ { 5 } 9 i$

Solve the problem. -How many different license plates can be made using 2 letters followed by 4 digits selected from the digits 0 through 9, if digits may be repeated but letters may not be repeated?

Solve the problem. -If a person puts 1 cent in a piggy bank on the first day, 2 cents on the second day, 3 cents on the third day, and so on, how much money will be in the bank after 60 days?

Evaluate the binomial coefficient. - $\left( \begin{array} { l } 4 \\4\end{array} \right)$

Solve the problem. -Suppose that $P ( A ) = 0.39 , P ( B ) = 0.36$ , and $P ( A \cap B ) = 0.21$ , find $P ( A \cup B )$ .

Solve the problem. -Of the coffee makers sold in an appliance store, 5% have either a faulty switch or a defective cord, 1.3% have a faulty switch, and 0.3% have both defects. What is the probability that a coffee maker will have a defective cord given that it has either a faulty switch or defective cord? Round to the nearest thousandth.

Solve the problem. -A brick staircase has a total of 19 steps The first step requires 111 bricks. Each successive step requires 4 fewer bricks than the prior one. How many bricks are required to build the staircase?

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $45 + 15 + 5 + \frac { 5 } { 3 } + \ldots$

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio. -6, -6, 6, -6, 6, . . .

Evaluate the binomial coefficient. - $\left( \begin{array} { l } 9 \\2\end{array} \right)$

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $- 6 - 2 - \frac { 2 } { 3 } - \ldots$

Write the first four terms of the sequence. - $a _ { n } = \frac { 3 n + 1 } { 3 n }$

Write a formula for the general term, or nth term, for the given sequence. -1 ·5, 2 ·6, 3 ·7, 4 ·8, 5 ·9, . . .