# Initialize Otter
import otter
grader = otter.Notebook("hw6-loops.ipynb")

🏡📝 Homework 6#

This assignment includes three problems on the topic of loops.

Question 1: Fibonacci Sequence

Fibonacci was born sometime around 1170 in the Kingdom of Pisa in what is now Italy. The sequences we now refer to as Fibonacci sequences first appeared in his treatise Liber Abaci from 1202 as a model of how rabbit populations grow.

A Fibonacci sequence is a list of integers in which the \(n\text{th}\) entry \(x_n\) is computed from:

\[ x_n = x_{n-1} + x_{n-2} \]

To initiate a Fibbonaci sequence, one must provide values for \(x_0\) and \(x_1\).

For example, if \(x_0 = 0\) and \(x_1 = 1\), the first 10 terms of the sequence are:

\[ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 \]

Your task is to implement a function that takes in two starting values and a length and returns the resulting Fibonacci sequence.

Write python code to do the following:

  • Define a function called fibonacci which accepts 3 input arguments: two starting values x0 and x1, and a sequence length L

  • Your function should return a list containing the first L fibonacci numbers starting with x0 and x1

  • You may assume the value L >= 2

Your code replaces the prompt: ...

...

grader.check("q1-fibonacci")

Question 2: Brute-Force Equation Solver

Numerous engineering and scientific applications require finding solutions to a set of equations. For example, the following system of equations:

\[ 8x + 7y = 38 \]
\[3x - 5y = -1 \]

have a solution: \( x = 3\), \(y = 2\).

You will implement a function that acts as a simple equation solver. For systems of equations of the form:

\[ ax + by = c \]
\[ dx + ey = f \]

You will use the following brute-force algorithm to find an integer solution for x and y in the range \([-10, 10]\) (if such a solution exists):

Notes:

  • You may assume that if a solution exists, there is only one solution. Your function should return the first solution found using the algorithm outlined above

  • Elegant mathematical techniques exist to solve such linear equations. However, for other kinds of equations or situations, brute-force approaches can be handy.

Write python code to do the following:

  • Define a function called eq_solver which accepts 6 input arguments, a, b, c, d, e, and f the coefficients for two equations

  • Your function should return a tuple of integers, containing solution values for (\(x\),\(y\))

  • If no solution was found, your function should return a string: "no solution found"

Your code replaces the prompt: ...

...

grader.check("q2-equation-solver")

Question 3: Convert to Binary

A natural number’s binary representation tell which powers of \(2\) sum to produce the number.

For instance: \(6\) in binary is \(110\), since: $\( 6 = (1)2^2 + (1)2^1 + (0)2^0 \)$

(\(110\) are the coefficients of the descending powers of \(2\).)

Your task is to write a function that takes in a positive integer and returns a string containing its binary representation. The following algorithm can be used to generate a natural number’s binary representation:

Reminders:

  • “x modulo 2”, written in python as x % 2, returns the remainder when dividing by two, either 0 or 1

  • Integer division written in python as, x // 2, truncates any remainder.

    • For example 5 // 2 = 2 since the remainder is truncated

  • Use str() to convert a number to a string

Write python code to do the following:

  • Define a function called to_binary that takes an input positive integer x

  • Using the algorithm described in pseudocode above, compute the binary representation of x

  • Return the binary as a string of "1"s and "0"s

Your code replaces the prompt: ...

...

grader.check("q3-binary")