Arithmetic vs Geometric Sequence: Key Differences
An arithmetic sequence adds the same constant (common difference d) to each term: 3, 7, 11, 15, … An geometric sequence multiplies each term by the same constant (common ratio r): 3, 6, 12, 24, … The distinction — adding vs multiplying — produces fundamentally different growth patterns: linear vs exponential.
This guide compares arithmetic and geometric sequences across formulas, graphs, sums, convergence, and real-world applications, with side-by-side examples to show when each model fits.
| Property | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Rule to next term | Add constant d: aₙ = aₙ₋₁ + d | Multiply by constant r: aₙ = aₙ₋₁ · r |
| nth term formula | aₙ = a₁ + (n − 1)d | aₙ = a₁ · rⁿ⁻¹ |
| Key parameter | Common difference d | Common ratio r = aₙ/aₙ₋₁ |
| Growth type | Linear (constant addition) | Exponential (constant multiplication) |
| Graph shape | Straight line through plotted points | Exponential curve (or decay if |r| < 1) |
| Sum of first n terms (Sₙ) | Sₙ = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n−1)d) | Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1 |
| Infinite sum | Diverges (unless d = 0) | Converges to a₁/(1−r) only if |r| < 1 |
| Middle term | Arithmetic mean: (a + c)/2 | Geometric mean: √(a · c) |
| Example | 2, 5, 8, 11, 14 … (d = 3) | 2, 6, 18, 54, 162 … (r = 3) |
| Real-world model | Equal monthly savings, uniform motion | Compound interest, population growth |
| Differences pattern | First differences are constant (= d) | Ratios of consecutive terms are constant (= r) |
| Negative terms | If d < 0 or a₁ < 0 | Alternate if r < 0, or all negative if a₁ < 0 and r > 0 |
Arithmetic Sequences — Linear Growth
An arithmetic sequence has the form a₁, a₁+d, a₁+2d, a₁+3d, … The common difference d is found by subtracting any term from the next: d = aₙ − aₙ₋₁. It is constant throughout the sequence.
nth term formula: aₙ = a₁ + (n − 1)d. The sequence grows linearly — plotting term index n vs term value aₙ gives a straight line with slope d.
Sum formula: Sₙ = n/2 · (a₁ + aₙ). This is Gauss's formula: pair first and last terms (their sum is constant), and there are n/2 such pairs. Equivalently, Sₙ = n/2 · (2a₁ + (n−1)d).
Example: Sequence: 5, 8, 11, 14, 17 … d = 3, a₁ = 5. Find a₁₀: a₁₀ = 5 + 9 · 3 = 32. Sum of first 10 terms: S₁₀ = 10/2 · (5 + 32) = 5 · 37 = 185.
Real-world uses: Equal monthly deposits into a savings account (without interest). Seat numbers in a row (1, 2, 3, …). Uniform motion at constant velocity (distance at each second). Temperature falling by 2°C per hour.
Geometric Sequences — Exponential Growth and Decay
A geometric sequence has the form a₁, a₁r, a₁r², a₁r³, … The common ratio r is found by dividing any term by the previous one: r = aₙ / aₙ₋₁. It is constant throughout.
nth term formula: aₙ = a₁ · rⁿ⁻¹. Plotting n vs aₙ gives an exponential curve — rising steeply if r > 1, decaying toward zero if 0 < r < 1, and alternating in sign if r < 0.
Sum formula: Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. When r = 1, all terms equal a₁ so Sₙ = n·a₁.
Infinite sum: If |r| < 1, the terms shrink to zero and the series converges: S∞ = a₁/(1 − r). If |r| ≥ 1, the series diverges. Example: 1 + 1/2 + 1/4 + … = 1/(1 − 0.5) = 2.
Real-world uses: Compound interest (each period multiplies the balance by 1+i). Radioactive decay (each half-life multiplies by 1/2). Population growth (each year multiplies by a growth factor). Depreciation (value multiplied by 0.8 each year → 20% depreciation). Virus spread (R₀ > 1 → geometric growth).
How to Identify Which Type You Have
From a table or list: Compute first differences (subtract consecutive terms). If they are constant → arithmetic. If not constant, compute ratios (divide consecutive terms). If they are constant → geometric. If neither is constant, it is neither type.
First difference test (arithmetic): 3, 7, 11, 15. Differences: 4, 4, 4 — constant. Arithmetic with d = 4.
Ratio test (geometric): 2, 6, 18, 54. Ratios: 3, 3, 3 — constant. Geometric with r = 3.
Mixed example: 1, 2, 4, 7, 11. Differences: 1, 2, 3, 4 — not constant. Ratios: 2, 2, 1.75, 1.57 — not constant. This is neither arithmetic nor geometric — it's a quadratic sequence (second differences: 1, 1, 1 — constant).
From a formula: Linear formula aₙ = 3n + 1 → arithmetic (d = 3). Exponential formula aₙ = 5 · 2ⁿ → geometric (r = 2). If the formula involves n as an exponent → geometric; n as a coefficient → arithmetic.
Sums and Series — Finite and Infinite
Arithmetic series (finite): Always converges to a finite value Sₙ = n/2 · (a₁ + aₙ). An infinite arithmetic series (with d ≠ 0) always diverges — the sum grows without bound. Example: 1 + 2 + 3 + 4 + … (arithmetic, d = 1) → diverges.
Geometric series (finite): Sₙ = a₁(1 − rⁿ)/(1 − r). Always finite for finite n. Example: 1 + 2 + 4 + … + 2⁹ = 1·(1 − 2¹⁰)/(1 − 2) = (−1023)/(−1) = 1023.
Geometric series (infinite): Converges only when |r| < 1. S∞ = a₁/(1 − r). This is used in loan payment derivations, present value of perpetuities, and probability (geometric distribution). Example: 8 + 4 + 2 + 1 + … = 8/(1 − 0.5) = 16.
Comparison: For the same first term and similar growth rates, a geometric series with r > 1 grows much faster than an arithmetic series. For large n, the geometric sum dominates. This is why compound interest (geometric) outperforms simple interest (arithmetic) over time.
Real-World Side-by-Side Examples
Savings with and without interest: Save $100/month. Without interest (arithmetic): after 12 months, total = 12/2 · (100 + 1200) = $7,800. With 1% monthly compound interest (geometric): each month your balance × 1.01. The geometric version gives slightly more — the difference grows over decades.
Drug dosage vs viral spread: Taking 200mg of medication daily (arithmetic addition, though real pharmacokinetics involves decay) vs a virus with R₀ = 2 (geometric doubling). After 10 periods: arithmetic = 2,000 mg total; geometric = 2¹⁰ = 1,024 cases. Geometric growth is exponentially more powerful.
Depreciation: Car loses $2,000/year (arithmetic) vs 15% per year (geometric). At year 5 on a $30,000 car: arithmetic value = $20,000; geometric value = $30,000 × 0.85⁵ ≈ $13,311. Real depreciation is geometric, making it much more severe than a flat annual loss suggests.
Algorithm complexity: An O(n) algorithm performs n steps (arithmetic-like linear growth). An O(2ⁿ) algorithm performs 2ⁿ steps (geometric). At n = 40: linear = 40 steps, exponential = 2⁴⁰ ≈ 1 trillion steps. This is why exponential-time algorithms are practically unusable for large inputs.
Verdict
Arithmetic sequences add a constant (linear growth, straight-line graph, always-diverging infinite sum). Geometric sequences multiply by a constant (exponential growth or decay, curved graph, convergent infinite sum when |r| < 1). The core test: constant differences → arithmetic; constant ratios → geometric.
- ✓To identify type: compute first differences (constant → arithmetic) or ratios (constant → geometric).
- ✓Arithmetic growth is linear — doubling the number of terms doubles the sum. Geometric growth is exponential — even moderate r > 1 produces enormous sums quickly.
- ✓Only geometric series can have a finite infinite sum — arithmetic series always diverge unless the common difference is zero.
- ✓The geometric mean √(ab) interpolates between two terms in a geometric sequence; the arithmetic mean (a+b)/2 does the same for arithmetic sequences.
- ✓In finance: simple interest is arithmetic; compound interest is geometric. Over long periods, the geometric (compound) version produces dramatically larger results.