[SOLVED] NYU Homework 6

Question 1:

The Fibonacci numbers sequence, Fn, is defined as follows:
F1 is1,F2 is1,andFn =Fn-1 +Fn-2
forn=3,4,5,…
In other words, each number is the sum of the previous two numbers. The first 10 numbers in Fibonacci sequence are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

Your program should interact with the user exactly as it shows in the following example:
Please enter a positive integer: 7
13

Question 2:

Your program should interact with the user to read the number of triangles in the tree and the character filling the tree.

Your implementation should include the following functions:
a. voidprintShiftedTriangle(intn,intm,charsymbol)

Write a program that, prints a ‘pine tree’ consisting of triangles of increasing sizes, filled

with a character (eg. ‘*’ or ’+’ or ‘$’ etc).

Left margin

It prints an n-line triangle, filled with symbol characters, shifted m spaces from the left margin.

For example, if we call printShiftedTriangle(3, 4, `+`), the expected output is:

+ +++

+++++

4 spaces

b. void printPineTree(int n, char symbol)
It prints a sequence of n triangles of increasing sizes (the smallest triangle is a 2-line triangle), which form the shape of a pine tree. The triangles are filled with the symbol character.

For example, if we call printPineTree(3, `#`), the expected output is:

        #
       ###
        #
       ###
      #####
        #
       ###
      #####
     #######

Left margin

Question 3:

The number e is an important mathematical constant that is the base of the natural logarithm. e also arises in the study of compound interest, and in many other applications.
Background of e: https://en.wikipedia.org/wiki/E_(mathematical_constant)

e can be calculated as the sum of the infinite series: 𝑒=1+!+!+!+!+⋯

!! #! $! %!

The value of e is approximately equal to 2.71828. We can get an approximate value of e, by calculating only a partial sum of the infinite sum above (the more addends we add, the better approximation we get).

Implement the function:

double eApprox(int n)

This function is given a positive integer n, and returns an approximation of e, calculated by the sum of the first (n+1) addends of the infinite sum above.

To test your function use the following main:

int main() { cout.precision(30);

for (int n = 1; n <= 15; n++) {

cout<<“n = “<<n<<‘\t'<<eApprox(n)<<endl; }

return 0; }

Notes:

1. Pay attention to the running time of eApprox. An efficient implementation would run in

   Θ(𝑛).

2. Since the values of the factorials will grow to be very large, use a variable of type double

   to store them.

Question 4:

a.

Implementation requirement: Pay attention to the running time of your function. An

efficient implementation would run in Θ%√𝑛𝑢𝑚).

b.

Question 5:

a. b.

Use the definition of Θ in order to show the following:

5n! + 2n” + 3n = Θ(𝑛!)

√7n” +2n−8= Θ(𝑛)