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 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 country uses as currency coins with values of 1 peso, 2 pesos, 5 pesos, and 10 pesos and bills with values of 5 pesos, 10 pesos, 20 pesos, 50 pesos, and 100 pesos. Find a recurrence relation for the number of ways to pay a bill of n pesos if the order in which the coins and bills are paid matters.

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 $n \geq 0, a_0 = 2, a_1 = 8$

A vending machine dispensing books of stamps accepts only one-dollar coins, $1 bills, and$5 bills. a) Find a recurrence relation for the number of ways to deposit n dollars in the vending machine, where the order in which the coins and bills are deposited matters. b) What are the initial conditions? c) How many ways are there to deposit $10 for a book of stamps?

a) Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s or two consecutive 1s. b) What are the initial conditions? c) How many ternary strings of length six do not contain two consecutive 0s or two consecutive 1s?

Find the number of positive integers not exceeding 1000 that are either the square or the cube of an integer.

Indica si lo que dice la oración es cierto o falso. Corrígela si es falso.

"Basta ya" es un eslogan que se usado en España en manifestaciones contra el terrorismo. Cierto ___ Falso ___

Contesta.

¿Qué es el presupuesto?

Choose the best response to each question.

$$\begin{array}{ll} \text{¿Cuándo es tu cumpleaños?}&\text{a. Es Juan.}\\ &\text{b. Mi cumpleaños es el dos de mayo.}\\ &\text{c. Es bajo y un poco perezoso.}\\ &\text{d. Soy tímido y serio.}\\ &\text{e. Tengo dieciséis años.}\\ &\text{f. Tiene quince años.}\\ \end{array}$$

Pati and Luis are looking at pictures of Pati's family. Complete their conversation with the correct Spanish possessive adjective.

LUIS: ¿Y quién es este señor?

PATI: ¿Con mi tía? Es (3) (her) esposo, Adolfo. Es el padrastro de mis primos.

Contesta.

¿Qué es la llama?

Ignacio y Dora hablan sobre las personas que ven en el campo. Combina las dos frases para formar una.

Modelo:

Ese chico es futbolista. Es mi amigo.

Ese chico, él que es futbolista, es mi amigo.

• Esos chicos corren en el campo de fútbol. Ellos son mis amigos.

La densidad del oro es $19.3\text{ g/cm}^3$.

(a) ¿Cuál es el volumen, en centímetros cúbicos, de una muestra de oro que tiene una masa de $0.715\text{ kg}$?

(b) Si esta muestra de oro es un cubo, ¿cuál es la longitud de cada borde en centímetros?