Before you turn this problem in, make sure everything runs as expected. First, restart the kernel (in the menubar, select Kernel$\rightarrow$Restart) and then run all cells (in the menubar, select Cell$\rightarrow$Run All).

Make sure you fill in any place that says YOUR CODE HERE or "YOUR ANSWER HERE", as well as your name and collaborators below:

In [ ]:
NAME = "Alyssa P. Hacker"
COLLABORATORS = "Ben Bitdiddle"

For this problem set, we'll be using the Jupyter notebook:


Part A (2 points)

Write a function that returns a list of numbers, such that $x_i=i^2$, for $1\leq i \leq n$. Make sure it handles the case where $n<1$ by raising a ValueError.

In [ ]:
def squares(n):
    """Compute the squares of numbers from 1 to n, such that the 
    ith element of the returned list equals i^2.
    
    """
    if n < 1:
        raise ValueError
    return [i ** 2 for i in range(1, n + 1)]

Your function should print [1, 4, 9, 16, 25, 36, 49, 64, 81, 100] for $n=10$. Check that it does:

In [ ]:
squares(10)
In [ ]:
"""Check that squares returns the correct output for several inputs"""
assert squares(1) == [1]
assert squares(2) == [1, 4]
assert squares(10) == [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
assert squares(11) == [1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121]
In [ ]:
"""Check that squares raises an error for invalid inputs"""
try:
    squares(0)
except ValueError:
    pass
else:
    raise AssertionError("did not raise")

try:
    squares(-4)
except ValueError:
    pass
else:
    raise AssertionError("did not raise")

Part B (1 point)

Using your squares function, write a function that computes the sum of the squares of the numbers from 1 to $n$. Your function should call the squares function -- it should NOT reimplement its functionality.

In [ ]:
def sum_of_squares(n):
    """Compute the sum of the squares of numbers from 1 to n."""
    return sum(squares(n))

The sum of squares from 1 to 10 should be 385. Verify that this is the answer you get:

In [ ]:
sum_of_squares(10)
In [ ]:
"""Check that sum_of_squares returns the correct answer for various inputs."""
assert sum_of_squares(1) == 1
assert sum_of_squares(2) == 5
assert sum_of_squares(10) == 385
assert sum_of_squares(11) == 506
In [ ]:
"""Check that sum_of_squares relies on squares."""
orig_squares = squares
del squares
try:
    sum_of_squares(1)
except NameError:
    pass
else:
    raise AssertionError("sum_of_squares does not use squares")
finally:
    squares = orig_squares

Part C (1 point)

Using LaTeX math notation, write out the equation that is implemented by your sum_of_squares function.

$\sum_{i=1}^n i^2$


Part D (2 points)

Find a usecase for your sum_of_squares function and implement that usecase in the cell below.

In [ ]:
import math

def hypotenuse(n):
    """Finds the hypotenuse of a right triangle with one side of length n and
    the other side of length n-1."""
    # find (n-1)**2 + n**2
    if (n < 2):
        raise ValueError("n must be >= 2")
    elif n == 2:
        sum1 = 5
        sum2 = 0
    else:
        sum1 = sum_of_squares(n)
        sum2 = sum_of_squares(n-2)
    return math.sqrt(sum1 - sum2)
In [ ]:
print(hypotenuse(2))
print(math.sqrt(2**2 + 1**2))
In [ ]:
print(hypotenuse(10))
print(math.sqrt(10**2 + 9**2))

Part E (4 points)

State the formulae for an arithmetic and geometric sum and verify them numerically for an example of your choice.

$\sum x^i = \frac{1}{1-x}$