Update Unit 2: Sequences, Series, and Financial Applications.md

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

@@ -100,7 +100,7 @@ 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 $`(x + y)^n`$
+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
@@ -111,3 +111,60 @@ A common relationship of binomial expansion is pascal's triangle. The $`nth`$ ro
       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})^n - 1]}{\frac{r}{n}}
+```