mirror of
https://gitlab.com/magicalsoup/Highschool.git
synced 2025-01-24 08:31:45 -05:00
178 lines
6.1 KiB
Markdown
178 lines
6.1 KiB
Markdown
# Unit 2: Sequences, Series, and Finicial Applications
|
|
|
|
|
|
## Terms
|
|
**sequence**: is an ordered set of numbres.
|
|
|
|
**Arithmetic Sequences**: is a sequence where the difference between each term is constant, and the constant is known as the `common difference`.
|
|
|
|
**Geometric Sequences**: is a sequence in which the ratio between each term is constant, and the constant is known as the `common ratio`.
|
|
|
|
**Note:** Not all sequences are arithmetic and geometric!
|
|
|
|
**finite series**: finite series have a **finite** number of terms.
|
|
- eg. $`1 + 2 + 3 + \cdots + 10`$.
|
|
|
|
**infinite series**: infinite series have **infinite** number of terms.
|
|
- eg. $`1 + 2 + 3 + \cdots`$
|
|
|
|
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.
|
|
|
|
**Series**: A series is the sum of the terms of a sequence.
|
|
|
|
|
|
## Recursion Formula
|
|
|
|
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.
|
|
|
|
1. Base term(s)
|
|
2. A formula to calculate each successive term.
|
|
|
|
eg. $`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \gt 1`$
|
|
|
|
## Aritmetic Sequences
|
|
|
|
Basically, you add the **commmon difference** to the current term to get the next term. As such, it follows the following pattern:
|
|
|
|
$`a, a+d, a+2d, a+3d, a+4d, \cdots`$. Where $`a`$ is the first term and $`d`$ is the **common difference**.
|
|
|
|
As such, the general term of the aritmetic sequence is:
|
|
|
|
$`\large t_n = a + (n - 1)d`$
|
|
|
|
## Geoemetric Sequences
|
|
|
|
Basically, you multiply by the **common ratio** to the current term toget the next term. As such, it follows the following pattern:
|
|
|
|
$`a, ar, ar^2, ar^3, ar^4, c\dots`$. Where $`a`$ is the first term and $`r`$ is the **common ratio**.
|
|
|
|
As such, the general term of the geometric sequence is:
|
|
|
|
$`\large t_n = a(r)^{n-1}`$
|
|
|
|
## Aritmetic Series
|
|
|
|
An arithmetic series is the sum of the aritmetic sequence's terms.
|
|
|
|
The formula to calculate is:
|
|
|
|
$`\large S_n = \dfrac{n(a_1 + a_n)}{2}`$ Or $`\large S_n = \dfrac{n(2a_1 + (n-1)d)}{2}`$
|
|
|
|
|
|
## Geometric Series
|
|
- A geoemtric series is created by adding the terms of the geometric sequence.
|
|
|
|
The formula to calulate the series is:
|
|
|
|
$`\large S_n= \dfrac{a(r^n- 1)}{r-1}`$ or $`\large S_n = \dfrac{a(1 - r^n)}{1 - r}`$
|
|
|
|
|
|
## Series and Sigma Notation
|
|
|
|
Its often convient to write summation of sequences using sigma notation. In greek, sigma means to sum.
|
|
|
|
eg. $`S_ = u_1 + u_2 + u_3 + u_4 + \cdots + u_n = \sum_{i=1}^{n}u_i`$
|
|
|
|
$`\sum_{i=1}^{n}u_i`$ means to add all the terms of $`u_i`$ from $`i=1`$ to $`i=n`$.
|
|
|
|
Programmers might refer to this as the `for` loop.
|
|
|
|
```cpp
|
|
int sum=0;
|
|
for(int i=1; i<=N; i++) {
|
|
sum += u[i];
|
|
}
|
|
```
|
|
|
|
## Infinite Geometric Series
|
|
|
|
Either the series **converges** and **diverges**. There is only a finite sum when the series **converges**.
|
|
|
|
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
|
|
series converges. Otherwise, $`r^n`$ goes to $`\infty`$, so the series diverges.
|
|
|
|
If the series diverges, then the sum can be calculated by the following formula:
|
|
|
|
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 $`|r| \lt 1`$
|
|
|
|
## Binomial Expansion
|
|
A binomial is a polynomial expression with 2 terms.
|
|
|
|
A binomial expansion takes the form of $`(x + y)^n`$, where $`n`$ is an integer and $`x, y`$ can be any number we want.
|
|
|
|
A common relationship of binomial expansion is pascal's triangle. The $`nth`$ row of the triangle correspond to the coefficents of $`(x + y)^n`$
|
|
|
|
```
|
|
1 row 0
|
|
1 1 row 1
|
|
1 2 1 row 2
|
|
1 3 3 1 row 3
|
|
1 4 6 4 1 row 4
|
|
1 5 10 10 5 1 row 5
|
|
```
|
|
|
|
The generalized version form of the binomial expansion is:
|
|
|
|
$`\large (x+y)^n = \binom{n}{0}x^ny^0 + \binom{n}{1} x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 + \cdots+ \binom{n}{n-1}x^{n-(n-1)}y^{n-1} + \binom{n}{0} x^0y^n`$.
|
|
|
|
Written in sigma notation, it is:
|
|
|
|
$`\large (x+y)^n = \sum_{k=0}^{n} \binom{n}{k}x^ky^{n-k}`$
|
|
|
|
eg. $`\large(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3`$
|
|
|
|
## Simple Interest
|
|
|
|
$`\large I = Prt`$
|
|
|
|
- $`P`$ is the principal money (start amount of $)
|
|
- $`r`$ is the annual interest rate expressed as a decimal (the percent is $`1 - r`$)
|
|
- $`t`$ is the time in years.
|
|
|
|
- This interest is calculated from the original amount each time. (eg. if you had $100, and your interest is 1\%, your interest will be a constant $1 each time.)
|
|
|
|
|
|
The total amount would be $`P + I`$.
|
|
|
|
## Compound Interest
|
|
|
|
Compound interest is interest paidon the interest previously earned and the original investment.
|
|
|
|
```math
|
|
\large A = P(1 + \frac{r}{n})^{nt}
|
|
```
|
|
|
|
- $`P`$ is the original amount
|
|
- $`\frac{r}{n} = i`$: this is the rate of interest **per period**.
|
|
- $`r`$ is interest rate
|
|
- $`n`$ is the number of periods (described below)
|
|
- $`nt`$ is the number of **total** periods (described below) Specifically, $`t`$ is the number of years.
|
|
- $`A`$ is the total value of the investment after $`nt`$ investemnt periods.
|
|
|
|
|Compounding Period|$`n`$|$`nt`$|
|
|
|:-----------------|:----|:-----|
|
|
|Annual|$`n = 1`$|$`nt = t`$|
|
|
|Semi-annual|$`n = 2`$|$`nt = 2t`$|
|
|
|Quarterly|$`n = 4`$|$`nt = 4t`$|
|
|
|Monthly|$`n = 12`$|$`nt = 12t`$|
|
|
|Daily|$`n = 365`$|$`nt = 365t`$|
|
|
|
|
## Future Value Annuities
|
|
**Definition:** An annuity is a series of equal deposits made at equal time intervales. Each depositis made at the end of each time interval.
|
|
|
|
A `Future Value` usually refers to how much money you will earn in the **future**. (eg. I have $100 dollars, I make desposits of $50 dollars each year with interest, how much will I have after $`5`$ years?)
|
|
|
|
Since it is basically the summation of a geometric sequence, we can apply the geometric series formula to get the following formula for future annuities:
|
|
|
|
```math
|
|
\large
|
|
FV = \frac{R[(1+\frac{r}{n})^{nt} - 1]}{\frac{r}{n}}
|
|
```
|
|
|
|
## Present Value Annuities
|
|
The **Present Value** of an annuity is today's value of having equally spaced payments or withdrawals of money sometime in the future.
|
|
|
|
```math
|
|
\large
|
|
PV = \frac{R[1 -(1+\frac{r}{n})^{-nt}]}{\frac{r}{n}}
|
|
``` |