Number and Algebra

For infinite geometric series where $|r|

< 1$, the sum to infinity is given by: $S_{\infty} = \frac{a_1}{1-r}$

Financial Applications

Geometric sequences and series have important applications in finance, particularly in compound interest calculations.

Laws of Exponents and Logarithms

Exponent Laws

1. $a^m \times a^n = a^{m+n}$

1. $\frac{a^m}{a^n} = a^{m-n}$

1. $(a^m)^n = a^{mn}$

1. $(ab)^n = a^n b^n$

1. $a^0 = 1$ (for $a \neq 0$)

1. $a^{-n} = \frac{1}{a^n}$

1. $a^{\frac{1}{n}} = \sqrt[n]{a}$

Logarithm Laws

1. $\log_a(xy) = \log_a x + \log_a y$

1. $\log_a(\frac{x}{y}) = \log_a x - \log_a y$

1. $\log_a(x^n) = n\log_a x$

1. $a^{\log_a x} = x$

1. $\log_a(a^x) = x$

Simple Deductive Proof

Deductive proof involves using logical reasoning to prove a statement. It typically follows these steps:

1. State the theorem or proposition to be proved

1. List the given information or assumptions

1. Use logical steps to deduce new information

1. Arrive at the desired conclusion

Binomial Theorem

The Binomial Theorem provides a formula for expanding $(x + y)^n$:

$$(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$$

where $\binom{n}{k}$ is the binomial coefficient, calculated as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Counting Principles (AHL)

Multiplication Principle

If one event can occur in $m$ ways, and another independent event can occur in $n$ ways, then the two events can occur together in $m \times n$ ways.

Permutations

A permutation is an arrangement of objects where order matters. The number of permutations of $n$ distinct objects is $n!$.

Combinations

A combination is a selection of objects where order doesn't matter. The number of ways to choose $r$ objects from $n$ objects is:

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

Complex Numbers (AHL)

Complex numbers take the form $a + bi$, where $i = \sqrt{-1}$.

Cartesian Form

In Cartesian form, complex numbers are written as $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Polar Form

In polar form, complex numbers are written as $z = r(\cos \theta + i \sin \theta)$, where $r$ is the modulus and $\theta$ is the argument.

Euler Form

Euler's formula connects exponential and trigonometric functions:

$e^{i\theta} = \cos \theta + i \sin \theta$

This leads to the Euler form of complex numbers: $z = re^{i\theta}$

De Moivre's Theorem (AHL)

De Moivre's theorem states that for any real number $x$ and integer $n$:

$(\cos x + i \sin x)^n = \cos(nx) + i \sin(nx)$

This theorem is particularly useful for finding powers of complex numbers and for solving certain trigonometric equations.

Proof Techniques (AHL)

Mathematical Induction

Induction is used to prove statements for all positive integers. It involves two steps:

1. Base case: Prove the statement for $n = 1$

1. Inductive step: Assume the statement is true for $n = k$, then prove it's true for $n = k+1$

Proof by Contradiction

This method assumes the opposite of what we want to prove, then shows this leads to a logical contradiction.

Proof by Counterexample

To disprove a universal statement, it's sufficient to find one counterexample where the statement doesn't hold.

Systems of Linear Equations (AHL)

Systems of linear equations can be solved using various methods:

1. Substitution

1. Elimination

1. Gaussian elimination

1. Matrix methods

Throughout the Math AA course, students are expected to develop skills in algebraic manipulation, problem-solving, and proof. The content builds from basic number operations to more advanced algebraic techniques and abstract concepts, providing a solid foundation for further mathematical study.