Number and Algebra

Sequence

Sequence

A list of numbers written in a specific order following a rule, where each number is called a term.

Series

The sum of the terms of a sequence, which can be finite or infinite.

Sigma Notation

A compact way to represent a series using the general term and the range of values the term takes.

Arithmetic Sequence

A sequence where the difference between consecutive terms is constant, with the nth term given by u_n = u_1 + (n-1)d.

Sum of Arithmetic Progression

The formula S_n = \frac{n}{2} (u_1 + u_n) = \frac{n}{2}[2u_1 + (n-1)d] gives the sum of the first n terms of an arithmetic sequence.

Geometric Sequence

A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Sum of Geometric Progression

The formula S_n = \frac{u_1(1-r^n)}{1-r} gives the sum of the first n terms of a geometric sequence, where r is the common ratio.

Proof

A logical argument that establishes the truth of a statement.

Direct Proof

A method of proof that involves constructing a series of reasoned connected facts to prove a statement.

Proof by Contradiction

A proof technique where the assumption that the statement is false leads to a contradiction.

Proof by Counterexample

A proof technique that shows a statement is false by providing a specific example that contradicts it.

Proof by Induction

A proof technique that demonstrates a statement is true for all positive integers by proving it for a base case and showing it holds for k+1 assuming it holds for k.