Quadratic and Non-Linear Sequences

Analyze the structural properties of real numbers, prove irrationality, explore set density, and convert recurring decimal expansions into exact rational forms.

Master integer and fractional exponent laws, convert between radical and exponential notation, and evaluate algebraic expressions with rational exponents.

Simplify surds, perform arithmetic operations with radical expressions, expand binomial surds, and rationalize single and binomial denominators.

Perform multi-step arithmetic operations with numbers in scientific notation and solve real-world scientific and astronomical scale problems.

Calculate bounds of rounded numbers and determine error propagation across multi-step arithmetic, geometric, and physical calculations.

Solve complex multi-part ratio problems, model direct and inverse proportion algebraically, and compute compound measures involving speed, density, and pressure.

Solve multi-step percentage change and reverse percentage problems, and model compound growth and depreciation over varying compounding intervals.

Analyze ordinary annuities, calculate regular payments and future values of sinking funds, model loan repayments, and compute progressive income taxes.

Derive explicit and recursive formulas for arithmetic sequences, calculate partial sums using sigma notation, and model linear progression phenomena.

Formulate explicit terms of geometric progressions, evaluate finite and infinite geometric series, and model exponential growth and annuity structures.

Analyze sequence difference patterns, derive nth-term quadratic formulas, and explore cubic, geometric, and recursive sequence models.