Chapter 8 - Binomial Expansion

How is Pascal’s triangle formed?

By adding adjacent pairs of numbers to generate the numbers in the next row.

What does the (n+1)th row of Pascal’s triangle represent?

It gives the coefficients in the expansion of (a+b)^n.

What is the definition of n factorial (n!)?

n! = n × (n−1) × (n−2) × … × 3 × 2 × 1.

How can factorial notation be used with Pascal’s triangle?

By calculating combinations using nCr = n!/(r!(n−r)!) to find entries quickly.

What is the binomial expansion of (a+b)^n?

(a+b)^n = a^n + (n choose 1)a^(n−1)b + (n choose 2)a^(n−2)b^2 + … + b^n.

What is the general term in the expansion of (a+b)^n?

The general term is (n choose r)a^(n−r)b^r.

How is the binomial expansion used for approximation?

If x is small, the first few terms can be used to approximate complicated expressions.