Understanding Arithmetic and Geometric Progressions
A sequence is an ordered set of numbers governed by a specific mathematical rule. Progressions are fundamental across number theory, calculus limits, financial amortization schedules, and geometric scaling.
Mathematical Formulas Summary
| Metric | Arithmetic Sequence (AP) | Geometric Sequence (GP) |
|---|---|---|
| Recursive Definition | $a_n = a_{n-1} + d$ | $a_n = a_{n-1} \cdot r$ |
| Explicit $n$-th Term ($a_n$) | $a_n = a_1 + (n - 1)d$ | $a_n = a_1 \cdot r^{n-1}$ |
| Sum of First $n$ Terms ($S_n$) | $S_n = \frac{n}{2}(2a_1 + (n - 1)d)$ | $S_n = a_1 \frac{1 - r^n}{1 - r} \quad (r \neq 1)$ |
| Sum to Infinity ($S_\infty$) | Diverges (no finite sum) | $S_\infty = \frac{a_1}{1 - r} \quad (\text{if } |r| < 1)$ |