1.2 Sequences and Series

Tags

Sequences

Series

Arithmetic

Geometric

Sigma

Compound interest

Finance

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Arithmetic sequences and series (1.2)

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Use of the formulae for the nthn^{th}nth term and the sum of the first nnn terms of the sequence.

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Use of sigma notations of sums of arithmetic sequences.

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Applications

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Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.

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Geometric sequences and series. (1.3)

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Use of the formulae for the nthn^{th}nth term and the sum of the first nn n terms of the sequence.

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Use of sigma notation for the sums of geometric sequences.

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Applications

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Financial applications of geometric sequences and series: ie) compound interest, annual depreciation. (1.4)

\begin{aligned} &\Delta \theta=\frac{\text { Arc }}{\text { Radius }}\\ &\Delta \boldsymbol{\theta}=\frac{\Delta \boldsymbol{s}}{r} \text { and } \boldsymbol{\Delta} \boldsymbol{\theta} \cdot \boldsymbol{r}=\boldsymbol{\Delta} \boldsymbol{s} \end{aligned}

Sequence

a_n

is an enumerated collection of objects in which repetitions are allowed and order matters.

There are two sequences to be aware of: arithmetic and geometric.

Sequences	Definition
Arithmetic	Arithmetic sequence has a common difference $d$ . General term is $a_n=a_0+(n-1)d$ . The sum is $k = \frac{1}{n} \sum_{k=1}^{n} a_k = \frac{n(\text{initial} + \text{final})}{2} = \frac{n(2a_0 + (n-1)d)}{2}$ . Other useful formulas include: 1. $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ 2. $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ 3. $\sum_{k=1}^{n}k^3=[\frac{n(n+1)}{2}]^2$
Geometric	Geometric sequence has a common ratio $r$ . General term is $a_n=a_0(r)^{n-1}$ . The sum is $\sum_{k=1}^{n} a_k = \frac{a_0(1-r^n)}{1-r}=\frac{a_0 (r^n-1)}{r - 1}$ . Also, for the sum until infinity, we have: $\sum_{n=1}^∞a_n=\frac{a_0}{1-r}$ .

These are used in different financial applications.

Applications	Definition
Compound interest	Compound interest means that every time interest is paid on an amount the added interest will also receive interest thereafter. The growth formula: $a_n=a_0[(1+\frac{r}{n})]^{Nt}$ Here, $r$ is the interest rate, $N$ the number of payments per year, and $t$ the number of years.