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

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

@@ -1,19 +1,19 @@
-# Unit 2: Sequences, Series, and Finicial Applications
+# Unit 2: Sequences, Series, and Financial Applications
 
 
 ## Terms
-**sequence**: is an ordered set of numbres.
+**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`.
+**Arithmetic Sequence**: 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`.
+**Geometric Sequence**: 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.
+**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.
+**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.
@@ -30,19 +30,19 @@ A sequence is defined recursively if you have to calculate a term in a sequence
 
 eg. $`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \gt 1`$
 
-## Aritmetic Sequences
+## Arithmetic Sequences
 
-Basically, you add the **commmon difference** to the current term to get the next term. As such, it follows the following pattern:
+Basically, you add the **common 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:
+As such, the general term of the arithmetic sequence is:
 
 $`\large t_n = a + (n - 1)d`$
 
-## Geoemetric Sequences
+## Geometric Sequences
 
-Basically, you multiply by the **common ratio** to the current term toget the next term. As such, it follows the following pattern:
+Multiply by the **common ratio** with the current term to get 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**.
 
@@ -50,7 +50,7 @@ As such, the general term of the geometric sequence is:
 
 $`\large t_n = a(r)^{n-1}`$
 
-## Aritmetic Series
+## Arithmetic Series
 
 An arithmetic series is the sum of the aritmetic sequence's terms.
 
@@ -60,16 +60,16 @@ $`\large S_n = \dfrac{n(a_1 + a_n)}{2}`$ Or $`\large S_n = \dfrac{n(2a_1 + (n-1)
 
 
 ## Geometric Series
-- A geoemtric series is created by adding the terms of the geometric sequence.
+- A geometric series is created by adding the terms of the geometric sequence.
 
-The formula to calulate the series is:
+The formula to calculate 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.
+It's often convenient to write summation of sequences using sigma notation. In Greek, sigma means to sum.
 
 eg. $`S_ = u_1 + u_2 + u_3 + u_4 + \cdots + u_n = \sum_{i=1}^{n}u_i`$
 
@@ -77,7 +77,7 @@ $`\sum_{i=1}^{n}u_i`$ means to add all the terms of $`u_i`$ from $`i=1`$ to $`i=
 
 Programmers might refer to this as the `for` loop.
 
-```cpp
+```java
 int sum=0;
 for(int i=1; i<=N; i++) {
     sum += u[i];
@@ -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 coefficents 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
@@ -125,7 +125,7 @@ eg. $`\large(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3`$
 
 $`\large I = Prt`$
 
-- $`P`$ is the principal money (start amount of $)
+- $`P`$ is the principal amount (start amount of $)
 - $`r`$ is the annual interest rate expressed as a decimal (the percent is $`1 - r`$)
 - $`t`$ is the time in years.
 
@@ -136,7 +136,7 @@ The total amount would be $`P + I`$.
 
 ## Compound Interest
 
-Compound interest is interest paidon the interest previously earned and the original investment. 
+Compound interest is interest paid on the interest previously earned and the original investment. 
 
 ```math
 \large A = P(1 + \frac{r}{n})^{nt}
@@ -158,11 +158,11 @@ Compound interest is interest paidon the interest previously earned and the orig
 |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.
+**Definition:** An annuity is a series of equal deposits made at equal time intervals. Each deposit is 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?)
+A `Future Value` usually refers to how much money you will earn in the **future**. (eg. I have $100, I make deposits 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:
+Since it is the summation of a geometric sequence, we can apply the geometric series formula to get the following formula for future annuities:
 
 ```math
 \large