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114 lines
3.8 KiB
Markdown
114 lines
3.8 KiB
Markdown
# Unit 2: Sequences, Series, and Finicial Applications
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## Terms
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**sequence**: is an ordered set of numbres.
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**Arithmetic Sequences**: is a sequence where the difference between each term is constant, and the constant is known as the `common difference`.
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**Geometric Sequences**: is a sequence in which the ratio between each term is constant, and the constant is known as the `common ratio`.
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**Note:** Not all sequences are arithmetic and geometric!
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**finite series**: finite series have a **finite** number of terms.
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- eg. $`1 + 2 + 3 + \cdots + 10`$.
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**infinite series**: infinite series have **infinite** number of terms.
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- eg. $`1 + 2 + 3 + \cdots`$
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Terms in a sequence are numbered with subscripts: $~t_1, t_2, t_3, \cdots t_n`$ where $`t_n`$is the general or $`n^{th}`$ term.
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**Series**: A series is the sum of the terms of a sequence.
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## Recursion Formula
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A sequence is defined recursively if you have to calculate a term in a sequence from previous terms. The recursion formula consist of 2 parts.
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1. Base term(s)
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2. A formula to calculate each successive term.
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eg. $`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \gt 1`$
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## Aritmetic Sequences
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Basically, you add the **commmon difference** to the current term to get the next term. As such, it follows the following pattern:
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$`a, a+d, a+2d, a+3d, a+4d, \cdots`$. Where $`a`$ is the first term and $`d`$ is the **common difference**.
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As such, the general term of the aritmetic sequence is:
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$`\large t_n = a + (n - 1)d`$
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## Geoemetric Sequences
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Basically, you multiply by the **common ratio** to the current term toget the next term. As such, it follows the following pattern:
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$`a, ar, ar^2, ar^3, ar^4, c\dots`$. Where $`a`$ is the first term and $`r`$ is the **common ratio**.
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As such, the general term of the geometric sequence is:
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$`\large t_n = a(r)^{n-1}`$
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## Aritmetic Series
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An arithmetic series is the sum of the aritmetic sequence's terms.
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The formula to calculate is:
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$`\large S_n = \dfrac{n(a_1 + a_n)}{2}`$ Or $`\large S_n = \dfrac{n(2a_1 + (n-1)d)}{2}`$
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## Geometric Series
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- A geoemtric series is created by adding the terms of the geometric sequence.
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The formula to calulate the series is:
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$`\large S_n= \dfrac{a(r^n- 1)}{r-1}`$ or $`\large S_n = \dfrac{a(1 - r^n)}{1 - r}`$
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## Series and Sigma Notation
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Its often convient to write summation of sequences using sigma notation. In greek, sigma means to sum.
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eg. $`S_ = u_1 + u_2 + u_3 + u_4 + \cdots + u_n = \sum_{i=1}^{n}u_i`$
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$`\sum_{i=1}^{n}u_i`$ means to add all the terms of $`u_i`$ from $`i=1`$ to $`i=n`$.
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Programmers might refer to this as the `for` loop.
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```cpp
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int sum=0;
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for(int i=1; i<=N; i++) {
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sum += u[i];
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}
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```
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## Infinite Geometric Series
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Either the series **converges** and **diverges**. There is only a finite sum when the series **converges**.
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Recall the our formula is $`\dfrac{a(r^n-1)}{r-1}`$, and is $`n`$ approaches $`\infty`$, if $`r`$ is less than $`1`$, then $`r^n`$ approaches $`0`$. So this
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series converges. Otherwise, $`r^n`$ goes to $`\infty`$, so the series diverges.
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If the series diverges, then the sum can be calculated by the following formula:
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If $`r = \dfrac{1}{2}`$, then $`\large \lim_{x \to \infty} (\frac{1}{2})^x = 0`$ Therefore, $`S_n = \dfrac{a(1 - 0)}{1 - r}`$. This works for any $`|r| \lt 1`$
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## Binomial Expansion
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A binomial is a polynomial expression with 2 terms.
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A binomial expansion takes the form of $`(x + y)^n`$, where $`n`$ is an integer and $`x, y`$ can be any number we want.
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A common relationship of binomial expansion is pascal's triangle. The $`nth`$ row of the triangle correspond to the coefficent of $`(x + y)^n`$
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```
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1 row 0
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1 1 row 1
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1 2 1 row 2
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1 3 3 1 row 3
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1 4 6 4 1 row 4
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1 5 10 10 5 1 row 5
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```
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