From 474a7e04d1bd8c17d1ca6905497a42571ac60bf2 Mon Sep 17 00:00:00 2001 From: Andrew Chen Date: Wed, 11 Mar 2020 16:28:03 +0000 Subject: [PATCH] Update Unit 2: Sequences, Series, and Financial Applications.md --- ...ces, Series, and Financial Applications.md | 44 +++++++++---------- 1 file changed, 22 insertions(+), 22 deletions(-) diff --git a/Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md b/Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md index 489c3cd..5e2ebcf 100644 --- a/Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md +++ b/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