Exam 12: Further Topics in Algebra

Provide an appropriate response. -Convert the decimal $.100100100 \ldots$ to a fraction by writing it as the infinite series $\frac { 100 } { 1000 } + \frac { 100 } { 1000 ^ { 2 } } + \frac { 100 } { 1000 ^ { 3 } } + \ldots$

Write the binomial expansion of the expression. - $( 5 x + 3 ) ^ { 4 }$

List the elements in the sample space of the experiment. -An 8-sided die is rolled. The sides contain the numbers $1,2,3,4,5,6,7,8$ . List the sample space of rolling one die.

Solve the problem. -Suppose that a family has 5 children and that the probability of having a girl is $\frac { 1 } { 2 }$ . What is the probability of having exactly two girls and three boys?

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $a _ { n } = \frac { 2 n } { n + 83 }$

Write the binomial expansion of the expression. - $\left( 2 x ^ { 2 } - y \right) ^ { 5 }$

List the elements in the sample space of the experiment. -A box contains 2 blue cards numbered 1 through 2, and 3 green cards numbered 1 through 3 . List the sample space of picking a blue card followed by a green card.

Solve the problem. -A baseball manager has 12 players of the same ability. How many 9 player starting lineups can he create?

Find the nth term of the geometric sequence. - $\frac { 1 } { 2 } , \frac { 1 } { 6 } , \frac { 1 } { 18 } , \ldots ; n = 6$

Evaluate the expression. - $\left( \begin{array} { c } 129 \\3\end{array} \right)$

Write the indicated term of the binomial expansion. - $( x + 3 y ) ^ { 9 ; } 8$ th term

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

Solve the problem. -How many ways can a class president, a class secretary, and a class treasurer be chosen from a group of 10 candidates?

Find a general term an for the geometric sequence. - $a _ { 2 } = - 4 , a _ { 11 } = \frac { 1 } { 128 }$

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. Round to the nearest thousandth, if necessary. -John does 9 pushups on the first day of a 30-day month, and then increases the number of pushups by 3 pushups a day. How many pushups has he done by the end of the month?

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $a _ { 16 } = - 67 , a _ { 17 } = - 6$

Solve the problem. -In how many ways can a class president and a class secretary be chosen from a group of 7 candidates?

Solve the problem. -A bag contains 6 apples and 4 oranges. If you select 5 pieces of fruit without looking, how many ways can you get 5 apples?

Evaluate the series, if it converges. - $6 + \frac { 24 } { 5 } + \frac { 96 } { 25 } + \frac { 384 } { 125 } + \ldots$

Use mathematical induction to prove that the statement is true for every positive integer n. - $\text { If } 0 < a < 1 \text {, then } a ^ { n } < a ^ { n - 1 } \text {. }$