# 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 coefficent 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        
```