Explains mathematical induction proof technique for sequences and statements indexed by natural numbers. Covers base case, inductive hypothesis, inductive step structure, and why induction works. Includes examples proving summation formulas and inequality statements with warnings about common errors.
Studies sequences growing at exponential rates including Fibonacci sequence. Presents multiply-shift-subtract method for geometric sums, characteristic root technique for solving recurrence relations, and handling repeated roots. Applies to tiling problems and exponential growth models.
Explores polynomial sequences with ﾎ婆-constant differences. Covers reverse-and-add technique for arithmetic sums, triangular numbers, polynomial fitting theorem, and finding closed formulas by fitting polynomials to sequence data. Includes chessboard squares counting problem.
Analyzes rate of growth in sequences. Defines arithmetic sequences (constant difference) with linear closed formulas and geometric sequences (constant ratio) with exponential formulas. Includes telescoping technique, partial sums, and iteration method for finding closed formulas.
Introduces sequences as ordered lists and functions, distinguishing closed formulas from recursive definitions. Uses Tower of Hanoi puzzle to illustrate sequence generation and pattern recognition. Explains sequence notation, indices, and multiple ways to describe sequences.