EXERCISE 3:

Building and using Pascal’s triangle:

Pascal’s triangle is used to obtain the coefficients for polynomial expansions: (x + y)^n

The triangle is:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

…

This means:

(x + y)^0 = **1**

(x + y)^1 = **1**x + **1**y

(x + y)^2 = **1**x^2
+ **2**xy + **1**y^2

(x + y)^3 = **1**x^3
+ **3** x^2 y + **3** xy^2 + **1**y^3

…

Note: each value is the addition of the values immediately
above and above-left. The first value at **(1,1)**
is **1**. Any unfilled value is** 0**. Thus, the element at **(2,1)** = **1** (above) + **0**(above-left)
= **1** and the element at **(2,2)** = **0** (above) + **1**
(above-left).

Question 1: Build the first 10 rows of Pascal’s triangle using for loops.

Hint: It is necessary to keep an extra columns of 0’s to the
left of the triangle, so that the **1**’s
are at position **(:,2)**.** **The above-left of position **(row, col)** is **(row –1, col –1)** and if **col**
is **1**, **col – 1** is **0**, which is
not a valid index in MATLAB.

Hint 2: The first element of your pascal triangle is at **(1,2)** and it is equal to **1**. The zeros are already there.

Hint 3: You 2 nested for loops, one for rows, one for columns.

Hint 4: After you finish, remove the extra column of 0’s

Question 2: Now that you have the triangle. Find the sum of each row and the sum of each column. What do they represent?

Question 3: (If you want the bonus marks): Remember Fibonacci’s numbers? They are also hidden in this triangle, can you find them? Hint, add diagonals.