# solution of practice exam
# here i use math library
# but you should have your own functions for ln and e

import math

# function Fibonacci(n) returns array
# of first n terms of Fibonacci sequence
# it is necessary for task 3
def Fibonacci(n):
    F = [0, 1]
    while len(F) < n:
        F.append(F[-1] + F[-2])
    return F

# you should have better function
# how many decimals of e do you remember?
def e():
    return 2.718281828

# here end functions that you should have ready
# from previous exercises
#####################################################

# task I

def taskI(n):
    P = [-1, 0]
    while len(P) < n:
        P.append(P[-1]**2 + 2*P[-1]*P[-2] + P[-2]**2)
    # now we need to take last iten in P
    # and get how many character it has
    # so we convert it to string and call len
    solution = len(str(P[-1]))
    print("Solution of task I is", solution)

# task II

def sinh(x):
    return (e()**x - e()**(-x)) / 2


def taskII(x):
    print("Solution of task II is:", round(sinh(math.log(x)), 3))
 

# task III

def taskIII(tol = 10**-9):
    n = 1
    while True:
        F = Fibonacci(n+1)
        new = F[-2]/F[-1]
        # a little trick, if n == 1,
        # you don't have to have old, becouse
        # if fails in the n > 1 part.
        if n > 1 and abs(old - new) < tol:
            break
        old = new
        n += 1
    print("Solution of task III is approx.:", new)
    
    
taskI(20)
taskII(1.618)
taskIII()