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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.
