**– Binomial expansion**

\(\dbinom{n}{r}\) = ^{n}C_{r }= \(\frac{n!}{(n! -r)! r!}\) and\(\dbinom{n}{n}\) = 1 and \(\dbinom{n}{1}\) = n

(a +bx)^{n} = \(\sum_{k = 0}^{k = n}\dbinom{n}{k}a^{n-k}bx^k\)

(x +1)^{n} = \(\dbinom{n}{0}x^n\) + \(\dbinom{n}{0}x^{n-1}\) +…..\(\dbinom{n}{n-1}x\) + \(\dbinom{n}{n}\)

If x=1

2^{n} = \(\dbinom{n}{0}\) +………..\(\dbinom{n}{n}\)

Pascal’s Triangle

If p > 1 and n > 1 \(\dbinom{n-1}{p-1}\) + \(\dbinom{n-1}{p}\) = \(\dbinom{n}{p}\)

**– Sequences defined as function of n**

Sequences can be defined as (1) explicitly u_{n } = f(n) or (2) by a recurrence relation u_{n+1} = f(u_{n })

Sequences can be decreasing , e.g (u_{n }) = \(\frac{1}{3n + 2}\) since u_{n+1} < u_{n}

Sequences can be increasing, e.g u_{n} = 3^{n} as u_{n+1} >u_{n}

Periodic sequence, e.g 0,2,0,2,0,….. has a period of 2

**– Sigma notation**

**– Arithmetic Sequences and Series**

u_{n } = u_{1} + (n – 1)d where d = common difference

d = u_{n+1} – u_{n}

S_{n} = \(\frac{(u_1 +u_n)n}{2}\) = \(\frac{n(2a +(n - 1)d}{2}\)

Arithmetic Mean of a and b, AM = \(\frac{a+b}{2}\)

**– Geometric Sequences and Series**

u_{n} = aq^{n-1} where q is the common ratio

q = \(\frac{u_{n+1}}{u_n}\)

S_{n} = \(\frac{a(1-q^n}{1 - q}\)

If |q|<1 then S_{n} = \(\frac{a}{(1 -q)}\)

Geometric Mean of a and b, GM = \(\sqrt{ab}\)

**-Harmonic Series**

A harmonic series is defined as S_{n} = \(\sum_{1}^{n}1/n\) (this is a divergent series) \(\lim_{n\to\infty}S_n\) = ∞

Harmonic Mean of a and b is HM = \(\frac{2}{\frac{1}{a}+\frac{1}{b}}\) = \(\frac{2ab}{a +b}\)

Root Mean Square of a and b, RMS = \(\sqrt{\frac{a^2 +b^2}{2}}\)

RMS ≥ AM ≥GM ≥HM

**– Sequences and Series in Modelling**

Ball bouncing on the floor (GP)

Investing in a compound interest earning account (GP)

Simple Interest (AP)