Ordered Lists
A Sequence is an ordered list of numbers following a specific rule. We call each number a "term" and use the notation \(a_n\) to represent the \(n\)-th term.
Types of Sequences
- Arithmetic: Add a constant amount to get the next term. Example: 2, 5, 8, 11... (Add 3).
- Geometric: Multiply by a constant amount to get the next term. Example: 3, 6, 12, 24... (Multiply by 2).
Worked Examples
Example 1: Finding the Rule
What is the rule for the sequence 10, 7, 4, 1...?
- The difference is \(7 - 10 = -3\).
- Formula: \(a_n = 10 + (n-1)(-3)\).
Example 2: Predicting the Future
Find the 100th term of the sequence \(a_n = 2^n\).
- \(a_{100} = 2^{100}\). (This is a massive number, best left in exponent form).
The Bridge to Quantum Mechanics
Quantum Mechanics is discrete. When an electron is in an atom, it can't have "any" energy; it can only have energies that fall into a specific Sequence. For example, the energy levels of a "Particle in a Box" follow the sequence \(E_n = n^2 \cdot E_1\). The energy levels of a Hydrogen atom follow the sequence \(E_n = -13.6/n^2\). To understand why atoms only emit specific colors of light, you must understand how to navigate these mathematical sequences. The "Quantum Numbers" you learn in chemistry are just indices (\(n\)) for these sequences.