Exam 12: Sequences, Series, and Probability

Solve the problem. -A traffic light is red for 40 seconds, yellow for 5 seconds, and green for 60 seconds. Find the probability: P(the light is green or yellow)

Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove the following statement for all positive integers n. - $2 + 5 + 8 + \ldots + ( 3 n - 1 ) = \frac { n } { 2 } ( 3 n + 1 )$

Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove the following statement for all positive integers n. - $1 + \frac { 5 } { 7 } + \frac { 25 } { 49 } + \ldots + \left( \frac { 5 } { 7 } \right) ^ { n - 1 } = \frac { 7 } { 2 } \left[ 1 - \left( \frac { 5 } { 7 } \right) ^ { n } \right]$

Evalutate the series. - $\sum _ { n = 1 } ^ { 4 } 2 ^ { n }$

Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove the following statement for all positive integers n. - $\left( \frac { 3 } { 4 } \right) ^ { n + 1 } < \frac { 3 } { 4 }$

Find the sum of the infinite geometric series. - $1 - \frac { 1 } { 4 } + \frac { 1 } { 16 } - \frac { 1 } { 64 } + \ldots$

Find the first and third partial sums, s1 and s3, for the sequence - $a _ { n } = \frac { n } { n + 2 }$

Evaluate the expression. - $\frac { 12 ! } { 9 ! }$

Find the sum of the terms in the geometric sequence. - $4 , - 1 , \frac { 1 } { 4 } , - \frac { 1 } { 16 } , \ldots$

Use the binomial theorem to expand the expression. - $\left( x ^ { 3 } + 2 y \right) ^ { 4 }$

Write the first five terms of the sequence whose nth term is shown. - $a _ { n } = 4 ( 4 n - 1 )$

Solve the problem. -A secured elevator allows access by pressing the correct sequence of letters on a control panel. If the control panel contains the 26 letters of the alphabet and a five-letter code must be entered (repetition is not permitted). How many different codes are possible?

Write out the series. - $\sum _ { n = 1 } ^ { 6 } ( 4 n - 3 )$

Solve the problem. -A pendulum swings through an arc 80 inches long on its first swing. Each swing thereafter, it swings only 70% as far as on the previous swing. How far will it swing altogether before coming to a complete stop? If necessary, Round to the nearest inch.

For the set of valueses $x _ { 1 } = 1 , x _ { 2 } = 3 , x _ { 3 } = - 2,$ 2, a nd x4 = 6, find the following. - $\sum _ { i = 1 } ^ { 4 } x _ { i } ^ { 2 } - 2$

For the given general term an, write an expression using $\Sigma$ to represent the indicated p to represent the indicated partial sum. - $a _ { n } = \frac { n } { n - 9 },$ third partial sum

Write the first five terms of the sequence whose nth term is shown. - $a _ { n } = 3 ^ { n }$

Find the indicated quantity of the arithmetic sequence. - $a _ { 1 } = 6 , a _ { n } = - 74 , d = - 4 ; \text { find } n$

Find the common difference of the sequence. - $\mathrm { a } _ { 1 } = 814 , \mathrm { a } _ { 4 } = 829 \text {; find } \mathrm { d }$

Solve the problem. -To save for retirement, Sue decides to deposit $2000 into an IRA at the end of each year for the next 30 years. If the interest rate is 5% per year compounded annually, find the value of the IRA after 30 years. (Round to the Nearest dollar.)