Fibonacci has done many things that have advanced mathematics. Most of his accomplishments came around 1200 A.D. since he had lived from c.1170 to c.1250. One of his most well known works is the Fibonacci Sequence. This sequence involves adding the latest two terms in the sequence in order to achieve the new term, written as F(n+1)=F(n)+F(n-1). This mathematical equation is not too hard to derive, especially given that the first two terms are defined to be 1 and 1 or 0 and 1 depending who you ask. There are other applications that add to this mathematical equation as well.
Using the Fibonacci Sequence, the Golden Spiral can be made. Using the Fibonacci numbers as guidelines, you can make squares that continually grow. Then, connecting corner to corner with an arc, a spiral can be made. The reason that this spiral has a special name, the Golden Spiral, is because in order to make this spiral, the dimensions are based off of the golden rectangle.
However, how would you answer the problem, how do the Fibonacci Sequence and Pascal's Triangle relate? Give an example such as 89 to help with understanding? After some research, you may find that the Fibonacci Sequence is made up of the additions of Pascal's Triangle's angles. Each line, always drawn parallel to the last, will make up the next number in the Fibonacci Sequence. So with Pascal's Triangle and the Fibonacci Sequence in hand, errors can be checked for on both. A good example is the number 89. 89 is the 11th number in the Fibonacci Sequence. This number is attained by adding 55 and 34 from the Fibonacci Sequence. Using Pascal's Triangle, 89 is attained by adding 1, 9, 28, 35, 15, and 1 (if you want to see more check here http://upload.wikimedia.org/wikipedia/commons/b/bf/PascalTriangleFibanacci.svg).