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.



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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:



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squares(10)



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"""Check that squares returns the correct output for several inputs"""
from nose.tools import assert_equal
assert_equal(squares(1), [1])
assert_equal(squares(2), [1, 4])
assert_equal(squares(10), [1, 4, 9, 16, 25, 36, 49, 64, 81, 100])
assert_equal(squares(11), [1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121])



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"""Check that squares raises an error for invalid inputs"""
from nose.tools import assert_raises
assert_raises(ValueError, squares, 0)
assert_raises(ValueError, squares, -4)

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.



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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:



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sum_of_squares(10)



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"""Check that sum_of_squares returns the correct answer for various inputs."""
assert_equal(sum_of_squares(1), 1)
assert_equal(sum_of_squares(2), 5)
assert_equal(sum_of_squares(10), 385)
assert_equal(sum_of_squares(11), 506)



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"""Check that sum_of_squares relies on squares."""
orig_squares = squares
del squares
try:
    assert_raises(NameError, sum_of_squares, 1)
except AssertionError:
    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.



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