Analyze arithmetic and geometric sequences. Find nth term, sum of first n terms, and visualize the progression.
A sequence is an ordered list of numbers following a specific rule. Two fundamental types in mathematics are arithmetic and geometric sequences. They appear in finance, physics, computer science, and daily life.
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Definition | Each term is obtained by adding a constant common difference (d) to the previous term. | Each term is obtained by multiplying the previous term by a constant common ratio (r). |
| Recursive Formula | aₙ = aₙ₋₁ + d | aₙ = aₙ₋₁ · r |
| Explicit Formula (nth term) | aₙ = a₁ + (n‑1)d | aₙ = a₁ · r⁽ⁿ⁻¹⁾ |
| Sum of First n Terms (Sₙ) | Sₙ = n/2 · (a₁ + aₙ) = n/2 · [2a₁ + (n‑1)d] |
Sₙ = a₁·(1 - rⁿ)/(1 - r) (r ≠ 1) If r = 1, Sₙ = n·a₁ |
| Infinite Sum (if exists) | Only converges if d = 0 (constant sequence); otherwise diverges. | Converges to a₁/(1‑r) only when |r| < 1. |
| Graph Shape | Points lie on a straight line (linear). | Points follow exponential growth/decay. |
| Common Examples | Monthly savings of $100: 100, 200, 300, … | Population doubling: 2, 4, 8, 16, … |
? Key Insight: The arithmetic sequence models constant rate of change, while the geometric sequence models constant relative (percentage) change.
If you start with a first term \(a_1\) and add the same value \(d\) repeatedly, you get an arithmetic progression. Example: \(3, 7, 11, 15, \dots\) has \(a_1 = 3\) and \(d = 4\). The 10th term is \(a_{10} = 3 + 9·4 = 39\). The sum of the first 10 terms can be computed quickly using the formula \(S_{10} = \frac{10}{2}(3 + 39) = 5·42 = 210\).
When you multiply by a fixed ratio \(r\), the sequence grows or decays exponentially. Example: \(2, 6, 18, 54, \dots\) has \(a_1 = 2\) and \(r = 3\). The 6th term is \(2·3^{5} = 2·243 = 486\). The sum of the first 6 terms is \(S_6 = 2·(1-3^6)/(1-3) = 2·(1-729)/(-2) = 728\).
If \(|r| < 1\), the terms become smaller and smaller, and the sum of infinitely many terms converges to a finite number: \[ S_{\infty} = \frac{a_1}{1 - r} \] For instance, \(100, 50, 25, 12.5, \dots\) with \(a_1=100, r=0.5\) has an infinite sum of \(100/(1-0.5) = 200\). This concept is crucial in physics (Zeno's paradox), finance (perpetuities), and signal processing.
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