Exam 11: Further Topics in Algebra

Use the summation properties to evaluate the series. The following rules may be needed: =; =; = =1 =1 =1 - $\sum _ { i = 1 } ^ { 5 } \left( - 2 i ^ { 2 } + 2 i - 6 \right)$

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

Answer the question. -Bruce agrees to go to work for Jim at a rate of $3 \notin$ for the first day, $6 \$$ for the second day, $12 \$$ for the third day, $24 \uparrow$ for the fourth day, and so on. What will Bruce's daily pay be on day 10 , and how much total wages will he have earned at the end of that day?

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest hundredth. - $\mathrm { S } 9 = - 243 \text {, a9 } = - 59$

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $\mathrm { a } _ { 3 } = - 18 , \mathrm {~d} = 4 , \mathrm { n } = 5$

Solve the problem. -Cal is packing his suitcase to go on a trip. He wants to pack 3 pairs of pants chosen from the 9 pairs of pants in his closet and 5 shirts chosen from the 10 shirts in his closet. In how many ways Can this be done?

It can be shown that $( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \frac { n ( n - 1 ) ( n - 2 ) } { 3 ! } x ^ { 3 } \ldots$ is true for any real number n (not just positive integer values) and any real number x, where $| x | < 1$ Use this series to approximate the given number to the nearest thousandth. - $4 + 4 \cdot \frac { 1 } { 7 } + 4 \cdot \left( \frac { 1 } { 7 } \right) ^ { 2 } + \ldots + 4 \cdot \left( \frac { 1 } { 7 } \right) ^ { n - 1 } = \frac { 4 \left( 1 - \left( \frac { 1 } { 7 } \right) ^ { n } \right) } { 1 - \frac { 1 } { 7 } }$

Evaluate the sum. - $\sum _ { k = 1 } ^ { 11 } ( 2 - 9 k )$

Solve the problem. -Suppose that a family has 5 children and that the probability of having a girl is $\frac { 1 } { 2 } \text {. What is the }$ probability of having no more than three boys?

Solve the problem. -Find the sum of the first 316 positive integers.

Find the nth term of the geometric sequence. - $4,8,16 , \ldots ; \mathrm { n } = 11$

Provide an appropriate response. -What is the common ratio in the geometric sequence $\left. a _ { n } = \log \left[ 6 ^ { n - 1 } \right) \right]$ ?

Solve the problem. -How many 10-digit telephone numbers (area code + number) are possible if the first digit cannot be zero, the first three digits cannot be 800 or 900, and the number must end in 0000?

Find the common ratio r for the given infinite geometric sequence. -80, 40, 20, 10, 5, . . .

Solve the problem. -If the police have 7 suspects, how many different sets of 5 suspects can they select for a lineup?

Provide an appropriate response. -Consider the selection of a nominating committee for a club. Is this a combination, a permutation, or neither?

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $\mathrm { a } _ { \mathrm { n } } = \frac { \mathrm { n } + 34 } { 2 \mathrm { n } }$

It can be shown that $( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \frac { n ( n - 1 ) ( n - 2 ) } { 3 ! } x ^ { 3 } \ldots$ is true for any real number n (not just positive integer values) and any real number x, where $| x | < 1$ Use this series to approximate the given number to the nearest thousandth. - $6 + 12 + 18 + \ldots + 6 n = 3 n ( n + 1 )$

Write the binomial expansion of the expression. - $( 3 x + 2 ) ^ { 3 }$

Solve the problem. -Find the sum of all the integers from 22 to 75 .