4,374
6,560
0
0
6
18
162
39,366
2,187
820
4,374
6,560
0
0
6
18
162
39,366
2,187
820
The Geometric Sequence Calculator computes any term, partial sums, and the infinite sum for a geometric sequence defined by its first term $$a_1$$ and common ratio $$r$$. A geometric sequence (or geometric progression, GP) is one where each term is obtained by multiplying the previous term by a fixed constant: $$a_1, a_1 r, a_1 r^2, a_1 r^3, \ldots$$
Geometric sequences are the discrete counterpart of exponential functions and appear throughout mathematics, science, and finance. The $$n$$-th term formula $$a_n = a_1 r^{n-1}$$ represents exponential growth when $$|r| > 1$$ and exponential decay when $$|r| < 1$$. When $$r < 0$$, the terms alternate in sign, creating oscillating patterns.
The partial sum formula $$S_n = a_1 \frac{1 - r^n}{1 - r}$$ (for $$r \neq 1$$) is one of the most important results in mathematics. It is derived by multiplying the sum by $$r$$ and subtracting, which telescopes all but the first and last terms. This technique was known to Euclid and appears in Book IX of the Elements.
Perhaps the most remarkable property of geometric sequences is that when $$|r| < 1$$, the infinite series $$\sum_{k=0}^{\infty} a_1 r^k = \frac{a_1}{1-r}$$ converges to a finite value. This is the geometric series formula, the foundation of countless applications: present value calculations in finance, probability distributions, signal processing, and fractal geometry.
In finance, geometric sequences model compound interest (balance after $$n$$ periods: $$P(1+r)^n$$), annuities, and mortgage payments. In physics, successive bounces of a ball (each reaching a fraction of the previous height) form a geometric sequence. In biology, cell division produces geometric growth. In computer science, binary search reduces the problem size geometrically, and geometric series bound the work of many divide-and-conquer algorithms.
Enter the first term $$a_1$$, common ratio $$r$$, and term number $$n$$. The calculator provides the $$n$$-th term, partial sum, infinite sum (when convergent), geometric mean, and several reference terms for a comprehensive analysis of your sequence.
Given first term $$a_1$$ and common ratio $$r$$:
n-th Term:
$$a_n = a_1 \cdot r^{n-1}$$
Partial Sum ($$r \neq 1$$): Start with $$S_n = a_1 + a_1 r + \cdots + a_1 r^{n-1}$$. Multiply by $$r$$: $$rS_n = a_1 r + a_1 r^2 + \cdots + a_1 r^n$$. Subtract:
$$S_n - rS_n = a_1 - a_1 r^n \implies S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$$
Infinite Sum ($$|r| < 1$$): As $$n \to \infty$$, $$r^n \to 0$$, so:
$$S_\infty = \frac{a_1}{1 - r}$$
Geometric Mean: The geometric mean of $$a_1$$ and $$a_n$$ is:
$$G = \sqrt{a_1 \cdot a_n} = a_1 \cdot r^{(n-1)/2}$$
This equals the middle term when $$n$$ is odd, and the interpolated midpoint when $$n$$ is even.
n-th Term (aₙ) gives the sequence value at position $$n$$. For $$|r| > 1$$, terms grow exponentially. For $$|r| < 1$$, terms shrink toward zero. For $$r < 0$$, terms alternate in sign.
Sum of First n Terms (Sₙ) is the partial sum. For $$|r| > 1$$, this grows exponentially with $$n$$. For $$|r| < 1$$, this approaches the infinite sum as $$n$$ increases.
Infinite Sum S∞ is computed only when $$|r| < 1$$ (convergent series). When $$|r| \geq 1$$, the series diverges and this displays 0 to indicate non-convergence. The infinite sum represents the total accumulated value if the sequence continued forever.
Geometric Mean of $$a_1$$ and $$a_n$$ is the value that would be the "midpoint" in a geometric sense — the square root of their product. In a geometric sequence, the geometric mean of any two terms equals the term halfway between them.
Inputs
Results
a₈ = 2·3⁷ = 2·2187 = 4374. Sum S₈ = 2·(1-3⁸)/(1-3) = 2·(1-6561)/(-2) = 6560. Since |r| = 3 > 1, the infinite sum diverges. The geometric mean of 2 and 4374 is √(2·4374) = √8748 ≈ 93.53.
Inputs
Results
Each term is half the previous: 100, 50, 25, 12.5, ... The 10th term is only 0.195. The partial sum S₁₀ = 199.8, very close to the infinite sum S∞ = 100/(1-0.5) = 200. After just 10 terms, 99.9% of the infinite sum is captured.
A geometric series converges if and only if |r| < 1. When convergent, the infinite sum is S∞ = a₁/(1-r). When |r| ≥ 1, the terms do not approach zero, so the series diverges. The boundary case |r| = 1 gives either a constant sequence (r=1) or an alternating sequence (r=-1), neither of which has a finite sum.
The common ratio r is the constant multiplier between consecutive terms: r = aₙ₊₁/aₙ. If r > 1, terms grow; if 0 < r < 1, terms shrink toward zero; if r < 0, terms alternate in sign. The ratio can be any real number except zero (which would make all terms after the first equal to zero).
Compound interest produces geometric sequences: balance = P·(1+r)ⁿ. The geometric series formula gives the future value of an annuity (regular deposits): FV = PMT·((1+r)ⁿ - 1)/r. Present value calculations use the infinite geometric series: PV = C/(r-g) for a perpetuity growing at rate g < r.
The geometric mean of two numbers a and b is √(ab). In a geometric sequence, any term is the geometric mean of the terms equally spaced around it. The geometric mean is the appropriate average for quantities that multiply (growth rates, ratios, indices), while the arithmetic mean is for quantities that add.
Yes. When r < 0, the terms alternate in sign: positive, negative, positive, ... For example, a₁ = 1, r = -2 gives 1, -2, 4, -8, 16, ... The convergence condition |r| < 1 still applies, so r = -0.5 gives a convergent alternating series.
When r = 1, all terms equal a₁ and the sequence is constant. The partial sum is simply S_n = n·a₁. The standard formula S_n = a₁(1-rⁿ)/(1-r) is undefined at r = 1 (0/0), so the calculator uses the special case S_n = n·a₁ directly.
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