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. - $S _ { n } : 1 + 6 + 11 + \ldots + ( 5 n - 4 ) = \frac { n ( 5 n - 3 ) } { 2 }$

The first several terms of a sequence are given. Find the indicated partial sum. - $- \frac { 1 } { 2 } , \frac { 1 } { 4 } , - \frac { 1 } { 8 } , \frac { 1 } { 16 } , \ldots ; S _ { 5 }$

Write a formula for the general term, or nth term, for the given sequence. - $\frac { 2 } { 3 } , \frac { 3 } { 4 } , \frac { 4 } { 5 } , \frac { 5 } { 6 } , \frac { 6 } { 7 } , \ldots$

Find the indicated sum. --3 + 1 + 5 + 9 + 13 + . . . + (4n - 7)

Solve the problem. -How many arrangements of answers are possible in a multiple-choice test with 9 questions, each of which has 5 possible answers?

Write a formula for the general term, or nth term, for the given sequence. --3, 9, -27, 81, . . .

Solve the problem. -Rewrite the number $0 . \overline { 285 }$ as the quotient of two integers.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $5 - \frac { 5 } { 3 } + \frac { 5 } { 9 } - \frac { 5 } { 27 } + \ldots$

Use the Binomial Theorem to expand the binomial. - $( \sqrt { x } + \sqrt { 5 } ) ^ { 4 }$

Find the indicated term of the sequence. --3, 9, -27, 81, . . .; a7

Solve the problem. -If you are dealt two cards successively (without replacement)from a standard 52-card deck, find the probability of getting a king on the first card and a queen on the second.

Write the first five terms of the geometric sequence with the given information. -The first term is 7 and the common ratio is $\frac { 1 } { 2 }$ .

Solve the problem. -If you are dealt two cards successively (with replacement of the first)from a standard 52-card deck, find the probability of getting a heart on the first card and a diamond on the second.

Rewrite the series using summation notation. Use 1 as the lower limit of summation. - $\frac { 1 } { 3 } + \frac { 1 } { 2 } + \frac { 3 } { 5 } + \ldots + \frac { 7 } { 8 }$

Solve the problem. -A card is drawn from a well-shuffled deck of 52 cards. What is the probability that the card will have a value of 3 and be a face card?

Find the sum of the geometric series. - $\sum _ { i = 1 } ^ { 10 } 400 ( 1.03 ) ^ { i } - 1 \quad$ Round the answer to four decimal places when necessary.

Use Pascal's triangle to expand the binomial. - $( x + 3 y ) ^ { 6 }$

Solve the problem. -If you are dealt two cards successively (without replacement)from a standard 52-card deck, find the probability of getting a black on the first card and a heart on the second.

Provide an appropriate response. -Find the 11th term of the geometric sequence whose first term is 5 and whose common ratio is 2.

Find the probability. -A bag contains 5 blue marbles, 5 green marbles, and 5 red marbles. One marble is drawn from the bag. What is the probability that the marble drawn is not blue?