## Problem 1: Fibonacci spiral

A Fibonacci spiral is formed by putting curved lines inside squares, whose size follows Fibonacci numbers, and are placed as follows This fractal is based on squares pattern below:

Write a function which takes a positive integer $$n$$ and returns the Fibonacci spiral up to that number.

Note that the spiral $$n$$ is obtained by rotating spiral $$n-1$$ -90 degrees (since it is clockwise rotation) and putting it aside a curve for $$n$$th square.

HINT: Please consult http://docs.racket-lang.org/teachpack/2htdpimage.html for using function add-curve to generate curved lines. An example of putting a curve on a -invisible/white- square of size 100 is as follows, replace 100 with the appropriate fibonacci number to obtain a useful shape:

(add-curve (square (fibonacciNumber n) "outline" "white")
1 100
90
0.4
100 1
0
0.4
"black")

## Problem 2: Population estimate

Natural logarithm of a number $$z$$ can be found using the following series:

$\ln (z) = 2\sum_{n=0}^\infty\frac{1}{2n+1}\left(\frac{z-1}{z+1}\right)^{2n+1}$

if the sum is limited at $$k$$ for practical computation: $\ln (z) = 2\sum_{n=0}^k\frac{1}{2n+1}\left(\frac{z-1}{z+1}\right)^{2n+1}$

Write a function to take $$z$$ and $$k$$ and return the approximated sum above, as its natural logarithm.

HINT: You can use the expt function to compute powers: e.g.(expt 2 3)` gives 8