# Recent questions in Engineering Mathematics

1
Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions that a plane is divided into by $n$ lines, if no two of the lines are parallel and no three of the lines go through the same point. Find $R_{n}$ using iteration.
2
A bus driver pays all tolls, using only nickels and dimes, by throwing one coin at a time into the mechanical toll collector. 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). In how many different ways can the driver pay a toll of $45$ cents?
3
Messages are transmitted over a communications channel using two signals. The transmittal of one signal requires $1$ microsecond, and the transmittal of the other signal requires $2$ microseconds. Find a recurrence relation for the number of different messages consisting of ... . What are the initial conditions? How many different messages can be sent in $10$ microseconds using these two signals?
4
Find a recurrence relation for the number of ternary strings of length $n$ that contain two consecutive symbols that are the same. What are the initial conditions? How many ternary strings of length six contain consecutive symbols that are the same?
5
Find a recurrence relation for the number of ternary strings of length $n$ that do not contain consecutive symbols that are the same. What are the initial conditions? How many ternary strings of length six do not contain consecutive symbols that are the same?
6
Find a recurrence relation for the number of ternary strings of length $n$ that contain either two consecutive $0s$ or two consecutive $1s.$ What are the initial conditions? How many ternary strings of length six contain two consecutive $0s$ or two consecutive $1s?$
7
Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive $0s$ or two consecutive $1s.$ What are the initial conditions? How many ternary strings of length six do not contain two consecutive $0s$ or two consecutive $1s?$
8
Find a recurrence relation for the number of ternary strings of length n that contain two consecutive $0s.$ What are the initial conditions? How many ternary strings of length six contain two consecutive $0s?$
9
A string that contains only $0s, 1s,$ and $2s$ is called a ternary string. Find a recurrence relation for the number of ternary strings of length $n$ that do not contain two consecutive $0s.$ What are the initial conditions? How many ternary strings of length six do not contain two consecutive $0s?$
10
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. What are the initial conditions? In many ways can this person climb a flight of eight stairs?
11
Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time. What are the initial conditions? In how many ways can this person climb a flight of eight stairs?
1 vote
12
Find a recurrence relation for the number of bit strings of length $n$ that contain the string $01$. What are the initial conditions? How many bit strings of length seven contain the string $01?$
13
Find a recurrence relation for the number of bit strings of length n that do not contain three consecutive $0s.$ What are the initial conditions? How many bit strings of length seven do not contain three consecutive $0s?$
14
Find a recurrence relation for the number of bit strings of length $n$ that contain three consecutive $0s.$ What are the initial conditions? How many bit strings of length seven contain three consecutive $0s?$
15
Find a recurrence relation for the number of bit strings of length $n$ that contain a pair of consecutive $0s$. What are the initial conditions? How many bit strings of length seven contain two consecutive $0s?$
16
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 $a_{1}, a_{2},\dots,a_{k},$ where $a_{1} = 1, a_{k} = n,$ ... What are the initial conditions? How many sequences of the type described in $(A)$ are there when $n$ is an integer with $n \geq 2?$
How many ways are there to pay a bill of $17$ pesos using the currency described in question $4,$ where the order in which coins and bills are paid matters?
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.
A vending machine dispensing books of stamps accepts only one-dollar coins, $\$1$bills, and$\$5$ bills. 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. What are the initial conditions? How many ways are there to deposit $\$10$for a book of stamps? 0 votes 0 answers 20 Find a recurrence relation for the number of permutations of a set with$n$elements. Use this recurrence relation to find the number of permutations of a set with$n\$ elements using iteration