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.

"""
### BEGIN SOLUTION
if n < 1:
raise ValueError("n must be greater than or equal to 1")
return [i ** 2 for i in range(1, n + 1)]
### END SOLUTION


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) == 
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."""
### BEGIN SOLUTION
return sum(squares(n))
### END SOLUTION


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 [ ]:
def pyramidal_number(n):
"""Returns the n^th pyramidal number"""
return sum_of_squares(n)


## Part E (4 points)¶

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