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Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md
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# Unit 2: Sequences, Series, and Finicial Applications # Unit 2: Sequences, Series, and Financial Applications
## Terms ## 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! **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. $`1 + 2 + 3 + \cdots + 10`$. - 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. $`1 + 2 + 3 + \cdots`$ - 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. 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. $`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \gt 1`$ 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**. $`a, a+d, a+2d, a+3d, a+4d, \cdots`$. 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:
$`\large t_n = a + (n - 1)d`$ $`\large t_n = a + (n - 1)d`$
## 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**. $`a, ar, ar^2, ar^3, ar^4, c\dots`$. 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:
$`\large t_n = a(r)^{n-1}`$ $`\large t_n = a(r)^{n-1}`$
## Aritmetic Series ## Arithmetic Series
An arithmetic series is the sum of the aritmetic sequence's terms. 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 ## 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 $`\large S_n = \dfrac{a(1 - r^n)}{1 - r}`$ $`\large S_n= \dfrac{a(r^n- 1)}{r-1}`$ or $`\large S_n = \dfrac{a(1 - r^n)}{1 - r}`$
## Series and Sigma Notation ## 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. $`S_ = u_1 + u_2 + u_3 + u_4 + \cdots + u_n = \sum_{i=1}^{n}u_i`$ 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. Programmers might refer to this as the `for` loop.
```cpp ```java
int sum=0; int sum=0;
for(int i=1; i<=N; i++) { for(int i=1; i<=N; i++) {
sum += u[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 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 1 row 0
@ -125,7 +125,7 @@ eg. $`\large(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3`$
$`\large I = Prt`$ $`\large I = Prt`$
- $`P`$ 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 $`1 - r`$) - $`r`$ is the annual interest rate expressed as a decimal (the percent is $`1 - r`$)
- $`t`$ is the time in years. - $`t`$ is the time in years.
@ -136,7 +136,7 @@ The total amount would be $`P + I`$.
## Compound Interest ## 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 ```math
\large A = P(1 + \frac{r}{n})^{nt} \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|$`n = 365`$|$`nt = 365t`$| |Daily|$`n = 365`$|$`nt = 365t`$|
## Future Value Annuities ## 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 ```math
\large \large