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

Unit 2: Sequences, Series, and Finicial Applications

Terms

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.

Geometric Sequences: 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. - eg. \(`1 + 2 + 3 + \cdots + 10`\).

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.

Series: A series is the sum of the terms of a sequence.

Recursion Formula

A sequence is defined recursively if you have to calculate a term in a sequence from previous terms. The recursion formula consist of 2 parts.

1. Base term(s)
2. A formula to calculate each successive term.

eg. \(`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \gt 1`\)

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.

int sum=0;
for(int i=1; i<=N; i++) {
    sum += u[i];
}

Infinite Geometric Series

Either the series converges and diverges. There is only a finite sum when the series converges.

Recall the our formula is \(`\dfrac{a(r^n-1)}{r-1}`\), and is \(`n`\) approaches \(`\infty`\), if \(`r`\) is less than \(`1`\), then \(`r^n`\) approaches \(`0`\). So this series converges. Otherwise, \(`r^n`\) goes to \(`\infty`\), so the series diverges.

If the series diverges, then the sum can be calculated by the following formula:

If \(`r = \dfrac{1}{2}`\), then \(`\large \lim_{x \to \infty} (\frac{1}{2})^x = 0`\) Therefore, \(`S_n = \dfrac{a(1 - 0)}{1 - r}`\). This works for any \(`|r| \lt 1`\)

Binomial Expansion

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 \(`(x + y)^n`\)

            1           row 0
           1 1          row 1
          1 2 1         row 2
         1 3 3 1        row 3
        1 4 6 4 1       row 4
      1 5 10 10 5 1     row 5