a) Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one, two, or three stairs at a time. b) What are the initial conditions? c) In many ways can this person climb a flight of eight stairs?

A bus driver pays all tolls, using only nickels and dimes, by throwing one coin at a time into the mechanical toll collector. a) Find a recurrence relation for the number of different ways the bus driver can pay a toll of n cents (where the order in which the coins are used matters). b) In how many different ways can the driver pay a toll of 45 cents?

a) Find a recurrence relation for the number of ways to completely cover a 2 × n checkerboard with 1 × 2 dominoes. b) What are the initial conditions for the recurrence relation in part (a)? c) How many ways are there to completely cover a 2 × 17 checkerboard with 1 × 2 dominoes?

Find a recurrence relation for the number of bit sequences of length n with an even number of 0s.

a) Find a recurrence relation for the number of ternary strings of length n that contain two consecutive 0s.

b) What are the initial conditions?

c) How many ternary strings of length six contain two consecutive 0s?

a) Find a recurrence relation for the number of bit strings of length n that do not contain three consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven do not contain three consecutive 0s?

a) Find a recurrence relation for the number of strictly increasing sequences of positive integers that have 1 as their first term and n as their last term, where n is a positive integer. That is, sequences a₁, a₂, . . . , aₖ, where a₁ = 1, aₖ = n, and aⱼ < aⱼ ₊ ₁ for j = 1, 2, . . . , k − 1.

a) Find a recurrence relation for the number of permutations of a set with n elements. b) Use this recurrence relation to find the number of permutations of a set with n elements using iteration.

Find the generating function for the finite sequence 2, 2, 2, 2, 2, 2.

Solve these recurrence relations together with the initial conditions given. a) $a_n = a_{n−1} + 6a_{n−2}$ for $n \geq 2$, $a_0 = 3, a_1 = 6$. b) $a_n = 7a_{n−1} − 10a_{n−2}$ for $n \geq 2$, $a_0 = 2, a_1 = 1.$c) $a_n = 6a_{n−1} − 8a_{n−2}$ for $n \geq 2, a_0 = 4, a_1 = 10$. d) $a_n = 2a_{n−1} − a_{n−2}$ for $n \geq 2, a_0 = 4, a_1 = 1$. e) $a_n = a_{n−2}$ for $n \geq 2, a_0 = 5, a_1 = −1$. f) $a_n = −6a_{n−1} − 9a_{n−2}$ for $n \geq 2, a_0 = 3, a_1 = −3$. g) $a_{n+2} = −4a_{n+1} + 5a_n$ for $n \geq 0, a_0 = 2, a_1 = 8$

Choose the word or phrase that is closest in meaning to the word in capital letters.

PROVISIONS
(A) food
(B) camping gear
(C) tools
(D) supplies

Choose the word or phrase that is closest in meaning to the word in capital letters.
MANEUVER

(A) hike
(B) feeling
(C) move
(D) cover-up

For each of the following sentences, underline the word or words in parentheses that correctly complete the sentence.

The legislature meets on the second floor of the (capital, capitol).

A small airline overbooks flights on the assumption that several passengers will not show up. Suppose that the probability that a passenger shows up is 0.91. What is the probability that a 20-seat flight with 22 tickets sold will be able to seat all passengers who arrive?