– 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
\(\sum_{k=1}^{k=n}1 = n\) \(\sum_{k=1}^{n}k = \frac{1}{2}n(n+1)\) \(\sum_{k=1}^{n}K^2 = \frac{1}{6}(n+1)(2n+1)\) \(\sum_{k=1}^{n}k^3 = \frac{1}{4}n^2(n+1)^2\)– 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)
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