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Concentration: Math AA HL

Algebra

Sequences and Series

Exponents and Logarithms

Functions

Polynimal Functions

Rational Functions

Exponential and Log Functions

Trigonometric Functions

Geometry and Trigonometry

Trigonometric Identities

2D and 3D Geometry

Vectors

Calculus

Limits and Continuity

Derivatives

Integrals

Statistics and Probability

Descriptive Statistics

Probability Concepts

Probability Distributions

Advanced Topics

Complex Numbers

Further Calculus

Matrices

Sequences and Series

An introduction to discrete patterns and progressions.

Introduction to the Concept

A sequence is an ordered list of numbers that follow a specific pattern. Each number in a sequence is called a term. We typically denote the terms of a sequence using subscript notation: $a_1, a_2, a_3, \dots, a_n$ .

A series is the sum of the terms of a sequence. If $\{ a_n \}$ is a sequence, then the corresponding series can be written with summation notation.

\sum\limits^{n}_{{i}={1}}{a_i}=a_1+a_2+a_3+\dots+a_n

Arithmetic Sequences

An arithmetic sequence is defined by a constant difference between consecutive terms. This is the common difference.

u_n = u_1 + (n-1)d

u_n

n-th term

u_1

first term

n

position

d

difference

INTUITION

To find any term, we start at $u_1$ and add the difference $d$ for every step taken.

u_1

u_1

u_2

u_1+d

u_3

u_1+d+d

u_4

u_1+3d

Notice that the number of differences added is always one less than the position of the term.

Example 1.1

Find the 15th term of the arithmetic sequence

3, 7, 11, \dots

u_1 = 3

d = 4

u_{15} = 3 + (15-1)4 = 59

Arithmetic Series

A series is the sum of the terms in a sequence. For an arithmetic progression, the sum of the first $n$ terms is given by two equivalent forms:

Last Term Known

S_n = \frac{n}{2}(u_1 + u_n)

Difference Known

S_n = \frac{n}{2}(2u_1 + (n-1)d)

INTUITION

We can write out the sum in two different ways

S_n=a_1+(a_1+d)+(a_1+2d)+\dots + (a_1+(n-1)d)

S_n=a_1+(a_1-d)+(a_1-2d)+\dots + (a_1-(n-1)d)