The Binomial Theorem, Fibonacci, and Vitruvian Dan

Vitruvian Dan by flickr user nogoodreason

Expand and simplify ...

a + b

Find a pattern, add two more rows to the triangle ...

11 1

1 2 1

1 3 3 1

1 4 6 4 1

Evaluate each term ...

Pascal's TriangleHow many different patterns can you find in the triangle?

Pascal's TriangleHow many different patterns can you find in the triangle?

http://mathworld.wolfram.com/TetrahedralNumber.html

Pascal's TriangleHow many different patterns can you find in the triangle?

Pascal's TriangleCan you find the Hockey Stick pattern?

Pascal's TriangleHow many different patterns can you find in the triangle?

1, 1, 2, 3, 5, 8, ....Fibonacci numbersCan you find them?

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

http://goldennumber.net/

The Binomial Theorem ...Algebraically

Combinatorically

Notice the patterns ...(1) The coefficient of the term is: (2) The exponent on a is given by: [n - (i - 1)](3) The exponent on b is given by: i - 1(4) This relation holds for each term in the expansion: [exponent on a] + [exponent on b] = n(5) The number of terms in any binomial expansion is: n + 1

Any individual term, let's say the ith term, in a binomial expansion can be represented like this:

Example: Find the 4th term in the expansion of

Determine the indicated term in each expansion.

(a) the 8th term in the expansion of

Determine the indicated term in each expansion.

(b) the 4th term in the expansion of

Determine the indicated term in each expansion.

(c) the middle term in the expansion of

40

right here

there

there is no middle term!!!

Example: Find the term that contains x in the expansion of 7

Example: Find the term that contains x in the expansion of 7

Find the term that contains in the expansion of: