# 9709. Pure 1. Series

Binomial Expansion

(a+b)n is called a binomial because it has two terms.

Consider the expansions (a+b)1, (a+b)2, (a+b)3, (a+b)4 (yes, let’s fully expand them and inspect the result). If you look at the coefficients of these do you notice any pattern?

Worked Examples

Exercise

Binomial coefficients

The numbers in Pascal’s triangle are actually the binomial coefficients $\binom nk=^nC_k=\frac{n!}{k!(n-k)!}$, where n is the row of Pascal’s triangle and k goes from 1 to n from left to right in the triangle.

We can calculate these manually, or using the calculator’s buttons ‘$^nC_k$‘ or ‘!’

In general, for all natural numbers n:

It is worth noticing that for all n, $\binom n0= 1$, $\binom n1= n$, $\binom nn= 1$ and for all n and k:

Worked Examples

Exercise

Arithmetic progression

• We use the following notation:
• a or a1 refers to the first term;
• d refers to the common difference;
• l referes to the last term.

Note that the common difference may be zero or a negative number.

The nth term is a + (n-1)d.

We use a method developed by Gauss to derive the formula for the sum of the first n terms in an arithmetic sequence. (Let’s do it!)

We should also be aware that the nth term in an arithmetic sequence is Sn-Sn-1

Worked Examples

Exercise

Geometric Progression

• We use the following notation:
• a or a1 refers to the first term;
• r refers to the common ratio;

The nth term is arn-1

We can multiply the whole sum by r and then subtract this to find the formula for the sum of n terms (Let’s do this!)

Worked Examples

Exercise

Infinite Geometric Progressions

If the absolute value of the common ratio is less than 1, then an infinite geometric series will converge.

S = $\frac{a}{1-r}$

Worked Examples

Exercise