Math

Geometric Progression: nth Term & Partial Sum

Free geometric progression calculator 2026: find nth term and partial sum instantly. Covers finance, physics, biology and CS uses. Calculate now.

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How to use this calculator

Follow this tool’s steps, then review its formula, assumptions, and limits below.

Step by step
01
Enter the first termType the value of a₁, the first term of the geometric progression. It can be any non-zero real number, including fractions or negatives.
02
Enter the common ratioType the value of r, the constant multiplicative factor between consecutive terms. Avoid r = 0; use values like 2, 0.5 or -1.5.
03
Enter the term numberType the integer n representing the position of the term you want to find or the number of terms to sum.
A geometric progression (GP) is one of mathematics' most powerful and ubiquitous structures: a sequence where every term is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio (r). Unlike arithmetic progressions that grow at a steady additive pace, geometric progressions scale exponentially — which is precisely why they appear everywhere from bank account balances and viral spread models to signal attenuation and algorithm analysis.

This calculator solves two fundamental outputs in a single step. First, the nth term: aₙ = a₁ · r^(n−1), which tells you the exact value at any position in the sequence without computing every prior term. Second, the partial sum of the first n terms: Sₙ = a₁ · (rⁿ − 1) / (r − 1) when r ≠ 1, or Sₙ = n · a₁ when r = 1. These two formulas together answer the two most common questions about any GP — 'what is the value at step n?' and 'what is the total accumulated across n steps?'

Why does this matter in practice? Consider a savings account growing at 8% annually: after 12 years, does it double? Triple? The GP formula gives you the precise answer in seconds without a spreadsheet. Or consider a disease spreading where each infected person infects 2.5 others per generation — by generation 10, how many total cases have accumulated? That's a GP partial sum. Engineers calculating echo decay in audio systems, biologists tracking bacterial colonies, and computer scientists analyzing recursive algorithms all reach for the same pair of equations this calculator implements.

What makes this tool distinct is that it handles edge cases cleanly — including r = 1 (constant sequence), fractional ratios (decaying sequences), negative ratios (alternating sequences), and very large exponents — while showing step-by-step intermediate values so you understand the computation, not just the answer. Whether you're a student checking homework, a financial analyst stress-testing a model, or an engineer verifying a decay curve, this calculator gives you immediate, accurate results with the context to use them correctly.

When to use this calculator

  • Compound Investment Growth —
  • Bacterial Colony Doubling —
  • Bouncing Ball Total Distance —
  • Merge Sort Memory Levels —
  • Radioactive Decay Remaining Mass —
  • Audio Echo Energy Decay —
  • Loan Amortization via Geometric Series —
  • Pyramid Sales Commission Structure —

nth Term (aₙ) and Partial Sum (Sₙ) for a₁ = 2 — Reference Values

rnaₙ = a₁ · r^(n−1)Sₙ
0.5122
0.530.53.5
0.550.1253.875
0.510≈ 0.003906≈ 3.996
1122
1326
15210
110220
2122
23814
253262
2101 0242 046
3122
331826
35162242
31039 36659 048
−1122
−1322 (n impar) / 0 (n par)
−1522 (n impar) / 0 (n par)
−11022 (n impar) / 0 (n par)

Fuente: fórmulas aₙ = a₁ · r^(n−1) y Sₙ = a₁ · (rⁿ − 1)/(r − 1) según NIST Digital Library of Mathematical Functions (dlmf.nist.gov). Valores calculados con a₁ = 2; para r = 1 se aplica Sₙ = n · a₁.

How it works

How It's Calculated

A geometric progression is fully defined by three parameters: the first term (a₁), the common ratio (r), and the number of terms (n).

nth Term Formula


aₙ = a₁ · r^(n − 1)

Partial Sum Formula


If r ≠ 1:   Sₙ = a₁ · (rⁿ − 1) / (r − 1)
If r = 1:   Sₙ = n · a₁
If |r| < 1 and n → ∞ (infinite GP):  S∞ = a₁ / (1 − r)

Worked Example (from calculator)


Given: a₁ = 2, r = 3, n = 5
a₅ = 2 · 3^(5−1) = 2 · 81 = 162
S₅ = 2 · (3⁵ − 1) / (3 − 1)
   = 2 · (243 − 1) / 2
   = 2 · 242 / 2
   = 242

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Reference Table

The table below shows aₙ and Sₙ for a₁ = 2 across different ratios and term numbers:

rn = 1n = 3 (aₙ)n = 3 (Sₙ)n = 5 (aₙ)n = 5 (Sₙ)n = 10 (aₙ)n = 10 (Sₙ)
0.520.53.50.1253.8750.00390625~3.996
1226210220
2281432621,0242,046
32182616224239,36659,048
1.0622.2476.3712.67611.2743.58226.362
−1222 or 0</td><td>2</td><td>2 or 022 or 0*

> *For r = −1, Sₙ alternates between a₁ (odd n) and 0 (even n).

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Typical Cases

Case 1 — Finance (Compound Interest)


A savings account starts at $500 with an annual growth rate of 8% (r = 1.08). What is the balance after 12 years (n = 12)?

a₁₂ = 500 · 1.08^11 = 500 · 2.3316 ≈ $1,165.80
S₁₂ = 500 · (1.08¹² − 1) / (1.08 − 1)
     = 500 · (2.5182 − 1) / 0.08
     ≈ $9,488.75  (total if you sum all yearly balances)

Case 2 — Biology (Bacterial Doubling)


Starting with 100 bacteria that double every hour (r = 2), how many are present after 8 hours (n = 9, counting the initial population as term 1)?

a₉ = 100 · 2^8 = 100 · 256 = 25,600 bacteria
S₉ = 100 · (2⁹ − 1) / (2 − 1) = 100 · 511 = 51,100 total bacteria-hours

Case 3 — Physics (Bouncing Ball)


A ball is dropped from 10 meters and rebounds to 60% of its height (r = 0.6). Total vertical distance after 6 bounces (n = 6 rebound terms, a₁ = 6 m for upward bounces):

S₆ = 6 · (1 − 0.6⁶) / (1 − 0.6)
   = 6 · (1 − 0.046656) / 0.4
   = 6 · 2.383 ≈ 14.30 m  (upward travel only)

Add the initial 10 m drop → total ≈ 38.60 m (counting both up and down).

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Common Mistakes

1. Off-by-one error on the exponent: The exponent is (n − 1), NOT n. Students frequently write aₙ = a₁ · rⁿ, which overshoots by one factor of r. For n = 1, a₁ · r⁰ = a₁ ✓, but a₁ · r¹ ≠ a₁ ✗.

2. Using the sum formula when r = 1: Plugging r = 1 into Sₙ = a₁ · (rⁿ − 1)/(r − 1) creates a division by zero. Always test r = 1 separately; the sum is simply n · a₁.

3. Confusing geometric with arithmetic progressions: In an arithmetic progression, you add a constant difference (d); in a GP, you multiply by a constant ratio (r). Mixing these leads to completely wrong results — e.g., calling r = 3 a "common difference" and computing a₅ = 2 + 4·3 = 14 instead of 162.

4. Applying the infinite-sum formula when |r| ≥ 1: The formula S∞ = a₁ / (1 − r) only converges when |r| < 1. Using it with r = 2 gives a negative "sum," which is mathematically meaningless for a divergent series.

5. Sign errors with negative ratios: When r is negative, aₙ alternates sign. For example, with a₁ = 4, r = −2: a₁ = 4, a₂ = −8, a₃ = 16. Forgetting the alternation leads to incorrect term signs and incorrect partial sums.

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    • Calculation Example

    a1=2, r=3, n=5
    a5=162, S5=242
    a5=162, S5=242

    Frequently asked questions

    What is the difference between a geometric progression and an arithmetic progression?
    In an arithmetic progression (AP), consecutive terms differ by a constant additive value called the common difference d. Example: 3, 7, 11, 15 has d = 4 — you always add 4. In a geometric progression (GP), consecutive terms differ by a constant multiplicative factor called the common ratio r. Example: 3, 6, 12, 24 has r = 2 — you always multiply by 2. The practical consequence is enormous: APs grow linearly (doubling n roughly doubles the nth term), while GPs grow exponentially (doubling n squares the nth term). This is why compound interest, population explosions, and viral spread are modeled with GPs, not APs. To test which type a sequence is, check whether consecutive differences are constant (AP) or consecutive ratios are constant (GP). A sequence can be neither if neither condition holds.
    What happens to the sum of a geometric series when |r| is less than 1?
    When the absolute value of the common ratio is strictly less than 1 (−1 < r < 1, r ≠ 0), each successive term is smaller than the previous one, shrinking toward zero. As a result, the infinite sum converges to a finite value given by the formula S∞ = a₁ / (1 − r). For example, with a₁ = 12 and r = 0.25, S∞ = 12 / 0.75 = 16 — the entire infinite series sums to exactly 16, even though it has infinitely many terms. This convergence property is foundational in physics (total distance of a bouncing ball), finance (present value of a perpetuity where payments shrink over time), and signal processing (total energy of decaying echoes). If |r| ≥ 1, the terms do not shrink and the series diverges — no finite infinite sum exists.
    Can the common ratio r be negative, zero, or a fraction?
    Zero is the only excluded value — r = 0 would collapse every term after the first to zero, making the sequence trivial and the ratio undefined (you'd be dividing by zero when computing aₙ₊₁/aₙ). Every other value is valid. A negative r produces an alternating sequence where terms flip between positive and negative signs each step: e.g., a₁ = 4, r = −2 gives 4, −8, 16, −32, 64. The standard formulas still apply, but you must track signs carefully — partial sums oscillate rather than monotonically increase. A fractional r between 0 and 1 (e.g., r = 0.5) produces a decreasing positive sequence converging toward zero; if it's between −1 and 0, you get a decreasing alternating sequence. Both cases use the same formulas, and if |r| < 1, the infinite-sum convergence formula S∞ = a₁/(1−r) also applies.
    How does the geometric progression formula relate to compound interest?
    Compound interest is the most widely applied real-world instance of a geometric progression. If you invest principal P at a periodic interest rate i (e.g., annual rate expressed as a decimal), the balance at the end of each period forms a GP: P, P(1+i), P(1+i)², P(1+i)³, … Here a₁ = P and r = (1+i). The balance after n periods is aₙ₊₁ = P · (1+i)ⁿ — the standard compound interest formula. The partial sum Sₙ applies when equal periodic deposits are made, yielding the future value of an ordinary annuity. For example, $2,500 invested at 5% annually for 15 years: a₁₆ = 2,500 · 1.05¹⁵ ≈ $5,197. Understanding this GP structure lets you deconstruct any compound-growth financial product — bonds, mortgages, index funds — into its mathematical fundamentals.
    How do I find the common ratio r if I only know two non-consecutive terms?
    If you know term aₘ at position m and term aₙ at position n (with n > m), the common ratio is: r = (aₙ / aₘ)^(1 / (n − m)). For example, if the 3rd term is 18 and the 7th term is 2,916: r = (2,916 / 18)^(1/4) = 162^(0.25) = 3. Once you have r, recover the first term with a₁ = aₘ / r^(m−1) = 18 / 3² = 2. This technique is essential in growth-rate estimation from sparse data — for instance, estimating a company's annual revenue growth rate given only revenues from two specific years separated by several years. It is also used in carbon dating calculations to find decay rates from two measured isotope concentrations.
    What is the geometric mean, and how is it connected to geometric progressions?
    The geometric mean of n positive numbers is the nth root of their product: G = (a₁ · a₂ · … · aₙ)^(1/n). In a GP, the geometric mean of all n terms equals the middle term (for odd n) or the square root of the product of the two middle terms (for even n). Example: in the 5-term GP 2, 6, 18, 54, 162, the geometric mean = (2 · 6 · 18 · 54 · 162)^(1/5) = (1,679,616)^0.2 = 18 = a₃. This property is directly applied in finance through the Compound Annual Growth Rate (CAGR): if a portfolio grows from $10,000 to $17,000 over 5 years, CAGR = (17,000/10,000)^(1/5) − 1 ≈ 11.2% — the geometric mean of the annual growth ratios. The geometric mean always understates the arithmetic mean for volatile data, making it a more conservative and accurate measure of average investment returns.
    How are geometric progressions used in computer science algorithms?
    GPs appear throughout algorithm analysis. In binary search, each comparison halves the remaining search space: n, n/2, n/4, …, 1 — a GP with r = 0.5 and n = log₂(N) terms, directly explaining the O(log N) time complexity. In merge sort, the number of sub-problems at recursion depth k is 2ᵏ — a GP with r = 2 — and the total work summed across all log₂(N) levels gives the O(N log N) complexity via a GP partial sum. Dynamic arrays (like Python lists or Java ArrayLists) double their capacity each time they run out of space, creating a GP of allocation sizes: 1, 2, 4, 8, 16, … The total memory allocated through k reallocations is S = 2ᵏ⁺¹ − 1, a GP partial sum, which proves amortized O(1) cost per insertion. Recognizing these GP structures is essential for algorithm design and complexity proofs.
    Is there a formula for the product (not sum) of the terms of a geometric progression?
    Yes. The product of the first n terms of a GP is Pₙ = a₁ⁿ · r^(n(n−1)/2). This follows from multiplying all terms: (a₁)(a₁r)(a₁r²)…(a₁r^(n−1)) = a₁ⁿ · r^(0+1+2+…+(n−1)) = a₁ⁿ · r^(n(n−1)/2). Example: for a₁ = 2, r = 3, n = 5: P₅ = 2⁵ · 3^10 = 32 · 59,049 = 1,889,568. A useful shortcut: for odd n, Pₙ = (middle term)ⁿ; for even n, Pₙ = (√(a₁ · aₙ))ⁿ. The product formula is used in statistics (computing geometric means efficiently), information theory (calculating joint probabilities of independent geometric processes), and number theory.
    What is the sum of an infinite geometric series, and when does it exist?
    An infinite geometric series a₁ + a₁r + a₁r² + a₁r³ + … converges to a finite sum only when |r| < 1. The formula is S∞ = a₁ / (1 − r). When |r| ≥ 1, the terms do not approach zero and the sum grows without bound (diverges). Practical examples: a ball dropped from 8 m rebounding to 60% each time has total downward travel S∞ = 8/(1−0.6) = 20 m (plus upward paths). A perpetuity paying $500 annually discounted at 5% has present value S∞ = 500/0.05 = $10,000. The convergence condition |r| < 1 is non-negotiable — applying the infinite sum formula when |r| ≥ 1 produces meaningless results. Always verify the ratio before using S∞.
    How do I identify whether a sequence is geometric from raw data?
    Divide each term by the preceding term. If the ratio aₙ₊₁/aₙ is identical for every consecutive pair, the sequence is geometric and that constant is r. Example: for the sequence 5, 15, 45, 135: 15/5 = 3, 45/15 = 3, 135/45 = 3 → confirmed GP with r = 3. If the ratios vary, the sequence is not a pure GP. In real-world data, ratios are rarely perfectly constant due to measurement noise. In that case, compute the geometric mean of all consecutive ratios as an estimated r, or use log-linear regression: taking logarithms of the terms converts a GP into an AP (log(aₙ) = log(a₁) + (n−1)·log(r)), and the slope of the best-fit line equals log(r). This log-linearization technique is standard in epidemiology, finance, and physics for fitting exponential growth models to noisy data.
    What are the most common mistakes when using the GP formula?
    Several errors recur frequently. First, off-by-one in the exponent: the nth term is a₁ · r^(n−1), not a₁ · rⁿ. For n = 1, a₁ = a₁ · r⁰ = a₁ — if you use rⁿ you get a₁ · r, which is actually a₂. Second, misidentifying a₁: ensure a₁ is truly the first term. If a problem gives you a₃ = 12 and r = 2, then a₁ = 12 / 2² = 3, not 12. Third, applying the r ≠ 1 sum formula when r = 1: Sₙ = a₁(rⁿ−1)/(r−1) is undefined when r = 1 (division by zero); the correct formula is Sₙ = n · a₁. Fourth, confusing periodic and annual rates in finance: if interest compounds monthly at 6% APR, r = 1 + 0.06/12 = 1.005 per month, not 1.06. Using the wrong r compounds errors exponentially. Fifth, forgetting to include the initial drop in bouncing-ball problems — the first downward path is not part of the rebound GP.
    How does a geometric progression relate to logarithms and exponential functions?
    A geometric progression is the discrete analog of a continuous exponential function. The nth term aₙ = a₁ · r^(n−1) has the same mathematical form as the exponential function f(x) = a · bˣ, where n plays the role of x and r plays the role of b. Taking the logarithm of both sides: log(aₙ) = log(a₁) + (n−1) · log(r), which is a linear equation in n — this is why plotting GP data on a logarithmic scale produces a straight line (log-linear or semi-log plot). This relationship is exploited constantly: epidemiologists plot case counts on log scales to spot exponential growth, engineers use Bode plots (log-log scale) to visualize frequency responses, and financial analysts use log-scale charts to compare percentage growth. Conversely, logarithms solve GP problems involving unknown n: if you need to find when aₙ reaches a target value T, solve n = 1 + log(T/a₁) / log(r).

    Methodology & trust

    Editorial

    Math calculator with its formula verified automatically against NIST Digital Library of Mathematical Functions — Combinatorial Analysis & Series (§26), per our editorial policy and methodology.

    Updates

    Updated: June 2026. Parameters are verified periodically against the cited sources.

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    Limitations

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    📌 How to cite this calculator
    APA format

    Rodríguez, M. (2026). Geometric Progression: nth Term & Partial Sum. Hacé Cuentas. https://hacecuentas.com/en/geometric-progression-sum-term

    BibTeX 
    @misc{hacecuentas_geometric_progression_sum_term_2026,
  author       = {Rodríguez, Martín},
  title        = {{Geometric Progression: nth Term & Partial Sum}},
  year         = {2026},
  howpublished = {\url{https://hacecuentas.com/en/geometric-progression-sum-term}},
  note         = {Hacé Cuentas}
}

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