Update Unit 2: Sequences, Series, and Financial Applications.md Not finished yet

Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md

@@ -18,6 +18,8 @@
 
 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
 
@@ -26,4 +28,60 @@ A sequence is defined recursively if you have to calculate a term in a sequence
 1. Base term(s)
 2. A formula to calculate each successive term.
 
-eg. $`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \ge 1`$
+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];
+}
+```
+
+