Mathematical Concepts Covered
Sequences
A sequence is an ordered list of numbers that follow a specific rule or pattern. Each number in the sequence is called a term.
Types of Sequences
1. Arithmetic Sequence: The difference between consecutive terms is constant (common difference ()).
• Example: 2, 4, 6, 8, 10… (Adding each time).
• Formula for the th term:
• Sum of first terms:
2. Geometric Sequence: Each term is obtained by multiplying the previous term by a common ratio .
• Example: 3, 6, 12, 24, 48… (Multiplying by ).
• Formula for the th term:
• Sum of first terms:
• Sum of an infinite geometric series (if ):
Polynomials, Polynomial Equations & Functions
Polynomial
A polynomial is an algebraic expression consisting of terms with variables, constants, and exponents, combined using addition, subtraction, and multiplication (but no division by variables).
• Example:
• Key Terms:
○ Degree: The highest exponent in the polynomial (e.g., has degree 2).
○ Leading Coefficient: The coefficient of the highest degree term (e.g., 3 in ).
○ Constant Term: A term without variables (e.g., 5 in ).
Polynomial Equations
A polynomial equation is formed when a polynomial is set equal to something, usually 0.
• Example:
Polynomial Functions
A polynomial function is a function where a polynomial is used to generate outputs based on an input .
• Example:
• Graphs:
○ The degree of the polynomial determines the shape of the graph.
○ Even-degree polynomials have symmetric, parabolic-like graphs.
○ Odd-degree polynomials have S-shaped graphs.
Concept of a Circle
A circle is a set of points that are all the same distance (called the radius) from a fixed center point.
Key properties of a circle:
• Radius (): The distance from the center to any point on the circle.
• Diameter (): The longest distance across the circle, passing through the center.
• Circumference: The perimeter (total length) of the circle.
• Area: The space inside the circle.
• Equation of a Circle: If the center is and the radius is , the equation is:
Combinatorics & Probability
Combinatorics:
Combinatorics is the branch of mathematics that deals with counting arrangements and selections efficiently.
• Permutation (): Arranging objects from , where order matters.
○ Example: How many ways can 3 books be arranged from 5?
• Combination (): Choosing objects from , where order does not matter.
○ Example: How many ways can 3 books be arranged from 5?
Probability
Probability: Probability measures how likely an event is to happen.
• Formula:
• Example:
○ Rolling a die and getting a 6:
○ Drawing an ace from a standard deck of 52 cards:
Types of Probability
1. Theoretical Probability: Based on mathematical reasoning.
• Example: The probability of flipping heads on a fair coin is
2. Experimental Probability: Based on actual experiments.
• Example: Flipping a coin 100 times and getting heads 48 times →
3. Conditional Probability: The probability of an event occurring given that another event has already occurred.
• Example: Probability of drawing a second ace after already drawing one ace from a deck.