Lesson 18 — Geometric progressions (GP)

Definition and formulas

General term

$a_n = a_1 q^{n-1}$

Sum of the first $n$ terms

For $q \neq 1$: $S_n = a_1 \cdot \frac{q^n - 1}{q - 1}$

Proof: $S_n = a_1 + a_1 q + \ldots + a_1 q^{n-1}$. Multiply by $q$: $q S_n = a_1 q + a_1 q^2 + \ldots + a_1 q^n$ Subtracting: $q S_n - S_n = a_1 q^n - a_1$, hence $S_n = a_1(q^n - 1)/(q - 1)$. ∎

Infinite sum (convergent GP)

If $|q| < 1$, $q^n \to 0$ as $n \to \infty$. Therefore:

$S_\infty = \lim_{n \to \infty} S_n = \frac{a_1}{1 - q}$

This is the geometric series, a centerpiece of Taylor series (Trim 9).

Behavior

• $q > 1$: GP grows exponentially.
• $q = 1$: constant.
• $0 < q < 1$: decreasing, converges to 0.
• $q = -1$: oscillates $a_1, -a_1, a_1, \ldots$
• $q < -1$: oscillates with growing amplitude.