Calculadora de progresión geométrica

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed nonzero number called the common ratio (r). Example: 2, 6, 18, 54, ... has common ratio 3. A geometric sequence is one of two foundational sequence types in algebra (the other is the arithmetic sequence, which uses addition instead of multiplication). This tutorial covers the nth term formula, the partial sum, the infinite sum, and how to identify and work with these sequences.

The nth term formula

If the first term of a geometric sequence is a and the common ratio is r, the nth term is:

aₙ = a × rⁿ⁻¹

Why the exponent is n − 1 and not n: by convention the first term has index n = 1, not 0. So a₁ = a × r⁰ = a × 1 = a. a₂ = a × r¹ = a × r. a₃ = a × r². The exponent is always one less than the term number.

Example: in 2, 6, 18, 54, ..., a = 2 and r = 3. The 7th term is a₇ = 2 × 3⁶ = 2 × 729 = 1458.

Identifying a geometric sequence

Given a list of numbers, divide consecutive pairs. If every quotient is the same, the sequence is geometric and that quotient is the common ratio r.

• 2, 6, 18, 54, ...: 6/2 = 3, 18/6 = 3, 54/18 = 3 ✓ geometric with r = 3.
• 1, 4, 9, 16, ...: 4/1 = 4, 9/4 = 2.25 ✗ NOT geometric. (This is the perfect squares — arithmetic on the differences but not on the ratios.)
• 100, 50, 25, 12.5, ...: 50/100 = 0.5, 25/50 = 0.5, 12.5/25 = 0.5 ✓ geometric with r = 0.5.

Sum of the first n terms (partial sum)

Adding the first n terms of a geometric sequence has a closed-form formula — you don't need to add term-by-term:

Sₙ = a × (1 − rⁿ) / (1 − r), valid when r ≠ 1.

If r = 1, every term equals a, so Sₙ = n × a (no formula needed).

Where the formula comes from: write the sum as S = a + ar + ar² + ... + arⁿ⁻¹. Multiply both sides by r: rS = ar + ar² + ... + arⁿ. Subtract: S − rS = a − arⁿ, so S(1 − r) = a(1 − rⁿ), giving S = a(1 − rⁿ)/(1 − r).

Example: sum of the first 6 terms of 5, 10, 20, 40, ...: a = 5, r = 2, n = 6. S₆ = 5(1 − 2⁶)/(1 − 2) = 5(1 − 64)/(−1) = 5(−63)/(−1) = 315.

Infinite sum — when the series converges

If |r| < 1 (the absolute value of the common ratio is strictly less than 1), the terms shrink toward zero and the infinite sum converges:

S∞ = a / (1 − r)

If |r| ≥ 1, the terms do not shrink and the sum diverges (grows without bound, or oscillates without settling).

Example: the infinite sum 1 + ½ + ¼ + ⅛ + ... has a = 1, r = ½, |r| < 1 ✓. S∞ = 1/(1 − ½) = 1/(½) = 2. This is the geometric series Zeno used in his "Achilles and the tortoise" paradox: each step covers half the remaining distance, so an infinite number of half-steps sum to a finite distance.

Another example: 1 + 2 + 4 + 8 + ... has r = 2, |r| = 2 > 1. Sum diverges to infinity — no finite value.

Recognizing and working with negative ratios

If r is negative, terms alternate sign: a, −a|r|, a|r|², −a|r|³, ...

Example: 3, −6, 12, −24, 48, ... has a = 3, r = −2.

Both the nth term formula and the sum formulas work as-is with negative r. The infinite sum converges if |r| < 1 even for negative r — e.g., 1 + (−½) + ¼ + (−⅛) + ... = 1/(1 − (−½)) = 1/(3/2) = 2/3.

Geometric mean — the multiplicative middle

The geometric mean of two positive numbers a and b is:

GM = √(a × b)

This is the middle term of a 3-term geometric sequence with first term a and last term b. For example, the geometric mean of 4 and 9 is √36 = 6 — and the sequence 4, 6, 9 has common ratio 1.5 throughout (6/4 = 9/6 = 1.5).

Geometric mean is preferred to arithmetic mean for averaging ratios, rates of return, growth factors, and other multiplicative quantities. The arithmetic mean of "doubled" and "tripled" gives 2.5×, but the geometric mean √(2 × 3) ≈ 2.45× is what compounds correctly.

Worked examples (full)

Example 1 — Find the 8th term: sequence 3, 6, 12, 24, ... has a = 3, r = 2. a₈ = 3 × 2⁷ = 3 × 128 = 384.

Example 2 — Solve for r given two terms: a₁ = 2 and a₅ = 162. Use the formula a₅ = a₁ × r⁴, so r⁴ = 162/2 = 81. Therefore r = ⁴√81 = 3 (taking the positive root; technically r could also be −3 with terms alternating sign, but the absolute value is uniquely determined).

Example 3 — Sum that diverges: the series 100 + 200 + 400 + ... + (term n) has a = 100, r = 2. The infinite sum diverges because |r| = 2 > 1. For any finite n, use Sₙ = 100(1 − 2ⁿ)/(1 − 2) = 100(2ⁿ − 1).

Real-world applications

• Compound interest. Balance after n compounding periods is B = P × (1 + i)ⁿ — a geometric sequence with a = P and r = (1 + i). The infinite-period sum diverges (money grows forever), so only partial sums are meaningful here.
• Population growth. Exponential growth models like P(t) = P₀ × eʳᵗ become geometric sequences when t is measured in discrete steps.
• Radioactive decay. Half-life decay is a geometric sequence with r = 1/2.
• Computer science. Doubling array sizes, binary tree heights, and geometric backoff in network protocols are all geometric sequences.
• Music. The frequencies of musical notes in a tempered scale form a geometric sequence with r = ¹²√2 ≈ 1.0595 per semitone.

Common mistakes

• Using n instead of n − 1 in the exponent. The first term is a × r⁰ = a, not a × r¹. The exponent is always one less than the term index.
• Applying the infinite sum formula when |r| ≥ 1. The series diverges. There is no finite "infinite sum" — partial sums grow without bound.
• Confusing geometric with arithmetic sequences. Arithmetic adds a fixed number each step; geometric multiplies by a fixed number. They are different sequence types with different formulas.
• Computing the geometric mean of negative numbers. GM = √(a × b) only makes sense when both a and b are non-negative (or when both are negative — taking the principal root of their product). Negative results are not defined for the geometric mean.
• Mixing the partial-sum and infinite-sum formulas. Sₙ = a(1 − rⁿ)/(1 − r) for finite n. S∞ = a/(1 − r) for the converging infinite case. They are not interchangeable.