6.1 KiB
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.
- Base term(s)
- 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 coefficents 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 The generalized version form of the binomial expansion is:
\(`\large (x+y)^n = \binom{n}{0}x^ny^0 + \binom{n}{1} x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 + \cdots+ \binom{n}{n-1}x^{n-(n-1)}y^{n-1} + \binom{n}{0} x^0y^n`\).
Written in sigma notation, it is:
\(`\large (x+y)^n = \sum_{k=0}^{n} \binom{n}{k}x^ky^{n-k}`\)
eg. \(`\large(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3`\)
Simple Interest
\(`\large I = Prt`\)
\(`P`\) is the principal money (start amount of $)
\(`r`\) is the annual interest rate expressed as a decimal (the percent is \(`1 - r`\))
\(`t`\) is the time in years.
This interest is calculated from the original amount each time. (eg. if you had $100, and your interest is 1%, your interest will be a constant $1 each time.)
The total amount would be \(`P + I`\).
Compound Interest
Compound interest is interest paidon the interest previously earned and the original investment.
\large A = P(1 + \frac{r}{n})^{nt}- \(`P`\) is the original amount
- \(`\frac{r}{n} = i`\): this is the
rate of interest per period.
- \(`r`\) is interest rate
- \(`n`\) is the number of periods (described below)
- \(`nt`\) is the number of total periods (described below) Specifically, \(`t`\) is the number of years.
- \(`A`\) is the total value of the investment after \(`nt`\) investemnt periods.
| Compounding Period | \(`n`\) | \(`nt`\) |
|---|---|---|
| Annual | \(`n = 1`\) | \(`nt = t`\) |
| Semi-annual | \(`n = 2`\) | \(`nt = 2t`\) |
| Quarterly | \(`n = 4`\) | \(`nt = 4t`\) |
| Monthly | \(`n = 12`\) | \(`nt = 12t`\) |
| 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.
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?)
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:
\large
FV = \frac{R[(1+\frac{r}{n})^{nt} - 1]}{\frac{r}{n}}Present Value Annuities
The Present Value of an annuity is today’s value of having equally spaced payments or withdrawals of money sometime in the future.
\large
PV = \frac{R[1 -(1+\frac{r}{n})^{-nt}]}{\frac{r}{n}}