Geometric Progression Calculator

Definition and Fundamental Properties

A geometric progression is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, 2, 6, 18, 54, 162 has a common ratio of 3. The sequence can be increasing (|r| > 1), decreasing (0 < |r| < 1), constant (r = 1), or alternating (r < 0). The general form is: a₁, a₁r, a₁r², a₁r³, a₁r⁴, ...

a_1, a_1r, a_1r^2, a_1r^3, \ldots, a_1r^n-1 \\ \textCommon ratio: r = \fraca_n+1a_n \text for all n

General Term Formula and Derivation

The nth term of a geometric progression can be calculated directly without computing all previous terms. Starting from a₁, the second term is a₁r, the third is a₁r², and by induction, the nth term is a₁r^(n-1). This formula is crucial for finding any term in the sequence efficiently. The formula works for any real number r ≠ 0 and can be extended to complex numbers.

a_n = a_1 × r^n-1 \\ \textwhere a_1 \text is the first term, r \text is the common ratio, n \text is the position

Sum of Finite Geometric Series

The sum of the first n terms of a geometric progression has a beautiful closed form. When r ≠ 1, we can derive the formula by multiplying the sum by r and subtracting. For r = 1, all terms equal a₁, so the sum is simply n·a₁. The formula Sₙ = a₁(1-rⁿ)/(1-r) is fundamental in mathematics and has countless applications in science, engineering, and finance.

S_n = \begincases (a_1(1-r^n))/(1-r) & \textif r ≠ 1 \\ n · a_1 & \textif r = 1 \endcases \\ \textAlternative form: S_n = (a_1(r^n-1))/(r-1) \text for r ≠ 1

Infinite Geometric Series and Convergence

An infinite geometric series converges to a finite sum when |r| < 1. As n approaches infinity, rⁿ approaches 0, giving the elegant formula S∞ = a₁/(1-r). This convergence is crucial in calculus and analysis. When |r| ≥ 1, the series diverges to infinity (if r > 1 or r = 1 with a₁ > 0) or oscillates without converging (if r ≤ -1). The convergence condition |r| < 1 means the terms are getting progressively smaller.

S_∞ = \lim_n \to ∞ S_n = (a_1)/(1-r) \text for |r| < 1 \\ \textConvergence: |r| < 1, \quad \textDivergence: |r| ≥ 1

Geometric Mean and Related Properties

In a geometric progression, any term (except the first and last) is the geometric mean of its two neighboring terms. That is, aₙ² = aₙ₋₁ × aₙ₊₁, or equivalently, aₙ = √(aₙ₋₁ × aₙ₊₁). This property uniquely characterizes geometric progressions. The geometric mean of all n terms is ⁿ√(a₁·a₂·...·aₙ) = a₁ⁿ·r^(n(n-1)/2)^(1/n).

a_n = √(a_n-1) × a_n+1 \\ \textGeometric Mean: GM_n = \sqrt[n]\prod_i=1^n a_i = a_1 · r^(n-1)/(2)

Relationship with Logarithms and Exponential Functions

Taking logarithms of terms in a geometric progression yields an arithmetic progression. If aₙ = a₁rⁿ⁻¹, then log(aₙ) = log(a₁) + (n-1)log(r), which forms an arithmetic sequence. This connection is fundamental in many applications. Conversely, if bₙ is an arithmetic sequence, then aₙ = e^(bₙ) forms a geometric progression. This relationship bridges linear and exponential growth.

\log(a_n) = \log(a_1) + (n-1)\log(r) \quad \text(Arithmetic Progression) \\ a_n = a_1 · r^n-1 = a_1 · e^(n-1)\ln(r)

Applications in Finance and Economics

Geometric progressions are essential in financial mathematics. Compound interest follows a geometric progression where each period's balance is multiplied by (1 + interest rate). The future value of an investment, annuities, perpetuities, and loan amortization all use geometric series. For example, with principal P and rate r, the amount after n periods is P(1+r)ⁿ. The present value of a perpetuity is calculated using the infinite GP sum formula.

\textCompound Interest: A = P(1+r)^n \\ \textAnnuity: PV = PMT · \frac1-(1+r)^-nr \\ \textPerpetuity: PV = (PMT)/(r)

Applications in Physics and Natural Sciences

Geometric progressions model exponential decay and growth in nature. Radioactive decay follows N(t) = N₀·e^(-λt), which discretely is a geometric sequence. Population growth (Malthusian model), bacterial growth, chemical reactions, and sound intensity (decibels) all involve geometric progressions. Newton's law of cooling, drug elimination from the body, and capacitor discharge in circuits follow geometric decay patterns.

\textRadioactive Decay: N(t) = N_0 · ((1)/(2))^t/t_1/2 \\ \textPopulation Growth: P(t) = P_0 · (1+r)^t \\ \textSound Intensity: L = 10\log_10((I)/(I_0)) \text dB

Geometric Progressions in Computer Science

Binary search algorithms divide the search space by 2 each iteration, forming a geometric sequence with r = 1/2. Merge sort and other divide-and-conquer algorithms have time complexities analyzed using geometric series. Hash table resizing typically doubles or halves capacity (geometric growth). Exponential backoff in network protocols uses geometric spacing between retry attempts. Memory allocation in dynamic arrays often grows geometrically.

\textBinary Search: T(n) = T(n/2) + O(1) = O(\log n) \\ \textGeometric Growth: 1, 2, 4, 8, 16, 32, \ldots \\ \textBackoff Time: t_n = t_0 · 2^n

Special Cases and Notable Examples

Powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...) are ubiquitous in computing. Powers of 10 (1, 10, 100, 1000, ...) form our decimal system. The geometric sequence 1, 1/2, 1/4, 1/8, ... sums to 2 (Zeno's paradox). In music, the chromatic scale uses r = ¹²√2 ≈ 1.059463. The Richter scale for earthquakes is logarithmic, related to geometric progressions.

\textPowers of 2: 2^0, 2^1, 2^2, 2^3, \ldots \\ \textZeno's Paradox: Σ_n=0^∞ (1)/(2^n) = 2 \\ \textMusical Note: f_n = f_0 · 2^n/12

Comparison with Arithmetic Progression

While arithmetic progressions have constant differences (aₙ - aₙ₋₁ = d), geometric progressions have constant ratios (aₙ/aₙ₋₁ = r). Arithmetic progressions grow linearly, geometric progressions grow exponentially. The sum of n terms in an AP is n(a₁+aₙ)/2, while in a GP it's a₁(1-rⁿ)/(1-r). Arithmetic means appear in APs between terms, geometric means in GPs. Both are fundamental in mathematics but model different types of change.

\textArithmetic: a_n = a_1 + (n-1)d, \quad S_n = (n(a_1+a_n))/(2) \\ \textGeometric: a_n = a_1r^n-1, \quad S_n = (a_1(1-r^n))/(1-r)

Worked Example

Inputs

• firstTerm: 2
• commonRatio: 3
• numberOfTerms: 4

Result: a₄ = 54, S₄ = 80

Explanation

Given a₁ = 2, r = 3, n = 4:

Step 1: Calculate the 4th term using aₙ = a₁ × r^(n-1) a₄ = 2 × 3^(4-1) = 2 × 3³ = 2 × 27 = 54

Step 2: Calculate the sum using Sₙ = a₁ × (1-r^n)/(1-r) S₄ = 2 × (1-3⁴)/(1-3) = 2 × (1-81)/(-2) = 2 × (-80)/(-2) = 80

The sequence is: 2, 6, 18, 54 Verification: Each term is 3 times the previous term (common ratio = 3)

Second Scenario

Inputs

• firstTerm: 2.4
• commonRatio: 3
• numberOfTerms: 4

Result: a₄ = 54, S₄ = 80

Explanation

This scenario uses different inputs (firstTerm = 2.4, commonRatio = 3, numberOfTerms = 4) to show how changing one variable affects the geometric progression result. Run the calculator above with these values to get the exact updated output with step-by-step work.