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 ff98b97..682c4e1 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 @@ -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. $`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \ge 1`$ \ No newline at end of file +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]; +} +``` + +