Exam 11: Further Topics in Algebra

Find the common ratio r for the given infinite geometric sequence. - $\frac { 4 } { 3 } , \frac { 16 } { 3 } , \frac { 64 } { 3 } , \frac { 256 } { 3 } , \frac { 1024 } { 3 } , \ldots$

Evaluate the sum. - $\sum _ { j = 5 } ^ { 18 } ( 2 j + 6 )$

Find a general term an for the geometric sequence. - $\mathrm { a } _ { 1 } = \frac { 1 } { 8 } , \mathrm { r } = 5$

Write the binomial expansion of the expression. - $\left( 2 + \frac { y } { 3 } \right) ^ { 5 }$

Solve the problem. -A game involves choosing 6 numbers from the numbers 1 through 13. In how many ways can this be done?

Solve the problem. -In how many ways can 9 people be chosen and arranged in a straight line, if there are 10 people from whom to choose?

Solve the problem. -John does 6 pushups on the first day of a 30 -day month, and then increases the number of pushups by 2 pushups a day. How many pushups has he done by the end of the month?

Find a general term an for the geometric sequence. - $2 , - \frac { 1 } { 2 } , \frac { 1 } { 8 } , - \frac { 1 } { 32 } , \frac { 1 } { 128 } \ldots$

Evaluate the sum using the given information. -x1 = 2, x2 = -3, x3 = -2, x4 = -5, and△x = -0.6; f(x) = x x- 5 4 ∑ f(xi)△x (Round to the nearest tenth, if necessary.) I=1

List the elements in the sample space of the experiment. -A box contains 3 blue cards numbered 1 through 3, and 4 green cards numbered 1 through 4. List the sample space of picking a blue card followed by a green card.

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. - $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$

Find the future value of the annuity. -To save for retirement, you decide to deposit $2500 into an IRA at the end of each year for the next 30 years. If the interest rate is 3% per year compounded annually, find the value of the IRA, Rounded to the nearest dollar, after 30 years.

Solve the problem. -Suppose there are 3 roads connecting town A to town B and 8 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

Provide an appropriate response. -Consider the selections of particular thirteen card bridge hands. Is this a combination, a permutation, or neither?

Use the formula for S_n to find the sum of the first five terms of the geometric sequence. - $\mathrm { a } _ { 1 } = - 3.449 , \mathrm { r } = 1.727$ (Round to the nearest hundredth)

Find a_n and a₆ for the following arithmetic sequence. - $\mathrm { a } _ { 19 } = 27 , \mathrm { a } _ { 21 } = 85$

Find the probability. -Two 6-sided dice are rolled. What is the probability the sum of the two numbers on the die will be 3?

Answer the question. -Bruce agrees to go to work for Jim at a rate of $1 \uparrow$ for the first day, $2 c$ for the second day, $4 c$ for the third day, $8 \$$ for the fourth day, and so on. What will Bruce's daily pay be on day 12 , and how much total wages will he have earned at the end of that day?

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 24 \left( \frac { 7 } { 2 } \right) ^ { i - 1 }$

Write out the first five terms of the sequence. - $a _ { n } = \frac { 4 } { n ^ { 2 } }$