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 [1]:
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 [2]:
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 [3]:
squares(10)

Out[3]:
[1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
In [4]:
"""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 [5]:
"""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 [6]:
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 [7]:
sum_of_squares(10)

Out[7]:
385
In [8]:
"""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 [9]:
"""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 [10]:
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 [11]:
print(hypotenuse(2))
print(math.sqrt(2**2 + 1**2))

2.23606797749979
2.23606797749979

In [12]:
print(hypotenuse(10))
print(math.sqrt(10**2 + 9**2))

13.45362404707371
13.45362404707371


## Part E (4 points)¶

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

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