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 = ""
COLLABORATORS = ""
```

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

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.
"""
# YOUR CODE HERE
raise NotImplementedError()
```

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")
```

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."""
# YOUR CODE HERE
raise NotImplementedError()
```

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
```

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

function.

YOUR ANSWER HERE

Find a usecase for your `sum_of_squares`

function and implement that usecase in the cell below.

In [ ]:

```
# YOUR CODE HERE
raise NotImplementedError()
```

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