When does the infinite geometric sum converge?

Only when |r| < 1 — the absolute value of the common ratio has to be less than 1, meaning each term is smaller than the last. Then the infinite sum is S∞ = a₁/(1 − r), a clean closed form. When |r| ≥ 1, the terms don't shrink to zero, and the sum diverges to ±∞. Zeno's paradox of Achilles and the tortoise is precisely a convergent geometric series — each step halves the remaining distance, but the infinite sum is finite.

Why is Gauss's trick so famous?

Story goes that Gauss, around age 8, was told by his teacher to sum 1+2+...+100 to keep the class busy. He figured out in seconds: pair first and last (1+100 = 101), second and second-to-last (2+99 = 101), and so on — 50 pairs of 101 equals 5,050. The general formula Sₙ = n/2 × (a₁ + aₙ) is the same trick generalized. Beautiful pattern recognition, and one of the most cited 'aha moments' in mathematics.

Can I have a negative common ratio?

Yes — and the terms alternate in sign. A geometric series with r = −2 starting at 1 gives 1, −2, 4, −8, 16, ... Such alternating series have surprising convergence properties — some converge that you wouldn't expect to, like the alternating harmonic series 1 − 1/2 + 1/3 − 1/4 + ... = ln(2). Calculus has whole chapters on when alternating series converge.

What about non-arithmetic, non-geometric sequences?

Plenty of other patterns exist — Fibonacci (each term is the sum of the previous two), factorials (n!), polynomial sequences, harmonic series. This calculator handles the two most common families because they cover most homework and finance applications. For Fibonacci-style recurrences, use the Scientific Calculator or write a quick program — it's just a two-line loop.

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Arithmetic & Geometric Sequence Sums

Math

Compute the n-th term and sum of arithmetic or geometric sequences. Term-by-term chart + cumulative growth line.

Math

Arithmetic & Geometric Sequence Sums

Generated on July 8, 2026

First term (a₁)

Common difference (d)

Number of terms (n)

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Arithmetic Sequence Result

a_10 = 29 · S_10 = 155

Generated

7/8/2026

a_10

Sum S_10

155

Mean per term

15.5

Individual terms

Cumulative sum

First 10 terms:

2, 5, 8, 11, 14, 17, 20, 23, 26, 29

Step-by-step calculation

Formula

Arithmetic: aₙ = a₁ + (n − 1)·d. Sₙ = n/2 · (2a₁ + (n − 1)·d) = n/2 · (a₁ + aₙ).

1Arithmetic sequence: a₁ = 2, d = 3, n = 10.
2n-th term: a_10 = 2 + (10 − 1) × 3 = 29.
3Sum: S_10 = 10/2 × (2·2 + (10 − 1)·3) = 155.
4Equivalently: S_10 = 10/2 × (2 + 29) = 155.

1 min readHow we build →

Quick Answer

Compute the n-th term and sum of arithmetic or geometric sequences. Term-by-term chart + cumulative growth line.

What the Arithmetic & Geometric Sequence Sums does

Math

The Sequence Sums Calculator computes the n-th term and partial sum of any arithmetic sequence (constant difference d, like 2, 5, 8, 11, …) or geometric sequence (constant ratio r, like 3, 6, 12, 24, …). For geometric series with |r| < 1, also reports the infinite sum (the limit as n → ∞). Includes a per-term bar chart showing how the terms grow, plus a cumulative-sum line chart showing how the running total accumulates.

Formula

Arithmetic: aₙ = a₁ + (n − 1)·d.   Sum Sₙ = n/2 · (a₁ + aₙ).   Geometric: aₙ = a₁ · r^(n−1).   Sum Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1.   Infinite geometric sum (|r| < 1): S∞ = a₁/(1 − r).

By Muhammad Salman SaleemMethodologyFree, no sign-up · Runs in your browser — your data stays private

What's worth knowing first

Personal finance scenario: you save Rs 10,000 every month with no interest — that's arithmetic. After 12 months you have Rs 120,000. Now consider Rs 10,000 invested at 8 percent annual compounding — that's geometric. After 30 years it's Rs 1,006,266 even without adding a single rupee. That's the gap between linear and exponential growth, and it's why financial planners drill 'start saving early' into clients. This calculator handles both: arithmetic sequences with constant difference d, geometric sequences with constant ratio r, and gives you both the n-th term and the partial sum Sₙ. For convergent geometric series (|r| < 1, where each term shrinks), it also gives you the infinite sum — the Zeno-paradox math where 1 + 1/2 + 1/4 + 1/8 + ... equals exactly 2. The per-term bar chart shows how the terms grow (or shrink), and the cumulative-sum line shows the running total accumulating. For arithmetic sequences, the cumulative line is a smooth parabola because partial sums grow quadratically. For geometric with r > 1, it's the famous hockey-stick curve. Worth knowing for exam day: the Gauss trick (1 + 2 + ... + 100 = 5,050) is the arithmetic-sum formula Sₙ = n/2 × (a₁ + aₙ) in action, and it's a beautiful piece of math that childhood Gauss reportedly figured out in seconds when his teacher tried to keep the class busy. Same trick, generalized to any arithmetic series.

Pro tips most people learn the hard way

Geometric beats arithmetic over the long run, every single time. Rs 10,000 invested at 8 percent grows to Rs 1 million in 30 years; Rs 10,000 stuffed under a mattress is still Rs 10,000 minus inflation. This is the single most important sequence-sum insight for personal finance — exponential compounding is genuinely magic over decades.
Gauss's trick — pair first and last to get a constant sum — generalizes to Sₙ = n/2 × (a₁ + aₙ) for any arithmetic series. This is one of those formulas worth genuinely memorizing because it shows up in exam questions about summing integers, salary projections, and saving patterns. Faster than computing term by term.
Convergent geometric series feel paradoxical the first time. 1 + 1/2 + 1/4 + 1/8 + ... = 2 exactly. Infinite terms, finite sum. This is the mathematical resolution to Zeno's paradox of Achilles never catching the tortoise — the infinite series of halved distances converges to a finite total, so Achilles does catch up.
For convergence in geometric series, the rule is strict: |r| < 1. Exactly |r| = 1 doesn't work (the terms don't shrink to zero, and the sum just keeps adding the same value). |r| > 1 obviously diverges. The boundary case is where infinite-series convergence problems get interesting in calculus.
Pro tip: if you're building a savings plan or modeling compound growth, the geometric series formula Sₙ = a₁(1 − rⁿ)/(1 − r) handles 'invest X today and watch it compound at rate r for n years.' If you add a fresh deposit each period, you need an annuity formula instead — which is actually arithmetic and geometric combined. Use the Compound Interest Calculator for that.

?What is the Arithmetic & Geometric Sequence Sums?

The Formula

Arithmetic: aₙ = a₁ + (n − 1)·d. Sum Sₙ = n/2 · (a₁ + aₙ). Geometric: aₙ = a₁ · r^(n−1). Sum Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. Infinite geometric sum (|r| < 1): S∞ = a₁/(1 − r).

Arithmetic sequences grow linearly — the bar chart shows a straight ascending line. Geometric sequences grow exponentially when |r| > 1 (each term multiplied by r) — the bar chart curves up steeply, showing the famous 'hockey stick' growth. When |r| < 1, geometric terms shrink toward zero and the infinite sum converges to a finite value (Zeno's paradox of Achilles and the tortoise is precisely this). When r = 1, every term equals a₁ and the sum is just n·a₁.

Practical Examples

Saving $100 every month with no interest: arithmetic with a₁=100, d=100, n=12 → year-end total $7,800.

Investment doubling each year: geometric with a₁=1, r=2 — by year 30 the term is 2^29 = 536,870,912.

Famous problem: 1 + 2 + 3 + … + 100 = 100/2 × (1+100) = 5,050. Gauss reportedly solved this at age 8.

Geometric series 1 + ½ + ¼ + ⅛ + … = 1/(1 − ½) = 2. Zeno's 'half-the-distance-each-step' paradox; the infinite sum is finite.

Loan amortization mathematically combines arithmetic + geometric series.

Powers of 10: 1 + 10 + 100 + ... + 10^6 = (10^7 − 1)/9 = 1,111,111.

Frequently Asked Questions

Arithmetic adds a constant (d) each step: 2, 5, 8, 11, ... (d = 3, you're adding 3 every time). Geometric multiplies by a constant (r) each step: 2, 6, 18, 54, ... (r = 3, you're tripling every time). Geometric grows exponentially while arithmetic grows linearly — and over a long enough horizon, geometric absolutely smokes arithmetic. The two patterns describe wildly different phenomena.

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Premium Converters

Arithmetic & Geometric Sequence Sums

Math

Compute the n-th term and sum of arithmetic or geometric sequences. Term-by-term chart + cumulative growth line.

Math

Arithmetic & Geometric Sequence Sums

Generated on July 8, 2026

First term (a₁)

Common difference (d)

Number of terms (n)

Save / share— download or share to WhatsApp, Facebook & more

Premium Converters

Arithmetic Sequence Result

a_10 = 29 · S_10 = 155

Generated

7/8/2026

a_10

Sum S_10

155

Mean per term

15.5

Individual terms

Cumulative sum

First 10 terms:

2, 5, 8, 11, 14, 17, 20, 23, 26, 29

Step-by-step calculation

Formula

Arithmetic: aₙ = a₁ + (n − 1)·d. Sₙ = n/2 · (2a₁ + (n − 1)·d) = n/2 · (a₁ + aₙ).

1Arithmetic sequence: a₁ = 2, d = 3, n = 10.
2n-th term: a_10 = 2 + (10 − 1) × 3 = 29.
3Sum: S_10 = 10/2 × (2·2 + (10 − 1)·3) = 155.
4Equivalently: S_10 = 10/2 × (2 + 29) = 155.

1 min readHow we build →

Quick Answer

Compute the n-th term and sum of arithmetic or geometric sequences. Term-by-term chart + cumulative growth line.

What the Arithmetic & Geometric Sequence Sums does

Math

Formula

Arithmetic: aₙ = a₁ + (n − 1)·d.   Sum Sₙ = n/2 · (a₁ + aₙ).   Geometric: aₙ = a₁ · r^(n−1).   Sum Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1.   Infinite geometric sum (|r| < 1): S∞ = a₁/(1 − r).

By Muhammad Salman SaleemMethodologyFree, no sign-up · Runs in your browser — your data stays private

What's worth knowing first

Pro tips most people learn the hard way

Geometric beats arithmetic over the long run, every single time. Rs 10,000 invested at 8 percent grows to Rs 1 million in 30 years; Rs 10,000 stuffed under a mattress is still Rs 10,000 minus inflation. This is the single most important sequence-sum insight for personal finance — exponential compounding is genuinely magic over decades.
Gauss's trick — pair first and last to get a constant sum — generalizes to Sₙ = n/2 × (a₁ + aₙ) for any arithmetic series. This is one of those formulas worth genuinely memorizing because it shows up in exam questions about summing integers, salary projections, and saving patterns. Faster than computing term by term.
Convergent geometric series feel paradoxical the first time. 1 + 1/2 + 1/4 + 1/8 + ... = 2 exactly. Infinite terms, finite sum. This is the mathematical resolution to Zeno's paradox of Achilles never catching the tortoise — the infinite series of halved distances converges to a finite total, so Achilles does catch up.
For convergence in geometric series, the rule is strict: |r| < 1. Exactly |r| = 1 doesn't work (the terms don't shrink to zero, and the sum just keeps adding the same value). |r| > 1 obviously diverges. The boundary case is where infinite-series convergence problems get interesting in calculus.
Pro tip: if you're building a savings plan or modeling compound growth, the geometric series formula Sₙ = a₁(1 − rⁿ)/(1 − r) handles 'invest X today and watch it compound at rate r for n years.' If you add a fresh deposit each period, you need an annuity formula instead — which is actually arithmetic and geometric combined. Use the Compound Interest Calculator for that.

?What is the Arithmetic & Geometric Sequence Sums?

The Formula

Practical Examples

Saving $100 every month with no interest: arithmetic with a₁=100, d=100, n=12 → year-end total $7,800.

Investment doubling each year: geometric with a₁=1, r=2 — by year 30 the term is 2^29 = 536,870,912.

Famous problem: 1 + 2 + 3 + … + 100 = 100/2 × (1+100) = 5,050. Gauss reportedly solved this at age 8.

Geometric series 1 + ½ + ¼ + ⅛ + … = 1/(1 − ½) = 2. Zeno's 'half-the-distance-each-step' paradox; the infinite sum is finite.

Loan amortization mathematically combines arithmetic + geometric series.

Powers of 10: 1 + 10 + 100 + ... + 10^6 = (10^7 − 1)/9 = 1,111,111.

Frequently Asked Questions

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

Browse all Math

GPA Calculator

Calculate your semester or cumulative Grade Point Average on the 4.0 scale, weighted by credit hours, with support for both weighted and unweighted grading systems.

Grade Calculator

Calculate your final grade from weighted categories like tests, homework, and exams.

Student Result Card Calculator

Subject-wise marks, grades, and a printable school result card — ideal for small schools and parents.

Fraction Calculator

Add, subtract, multiply, and divide proper, improper, and mixed fractions with automatic reduction to lowest terms and decimal equivalents for cooking, carpentry, and homework.

Ratio Calculator

Simplify ratios to lowest terms using GCD and solve proportions like a:b = c:x for any missing value — useful for recipes, map scales, aspect ratios, and mixing ingredients.

Average Calculator

Compute the arithmetic mean, median, mode, range, count, sum, minimum, and maximum from any list of numbers — essential for descriptive statistics, surveys, and grading.

LCM / GCD Calculator

Find the LCM and GCD (HCF) of two or more integers using the Euclidean algorithm — useful for simplifying fractions, finding common denominators, and scheduling repeating events.

Prime Number Checker

Test whether any integer is a prime number using trial division up to its square root, and view the complete unique prime factorization for composite numbers.

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See every tool in this category, plus FAQs and category-specific guides.

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Islamic9 Financial14 Gold & Silver8 Health25 Unit19 Date & Time9 Construction22 Utility20 Fun12

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Arithmetic & Geometric Sequence Sums

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What the Arithmetic & Geometric Sequence Sums does

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Pro tips most people learn the hard way

?What is the Arithmetic & Geometric Sequence Sums?

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Practical Examples

Frequently Asked Questions

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Arithmetic & Geometric Sequence Sums

Arithmetic & Geometric Sequence Sums

Arithmetic Sequence Result

Individual terms

Cumulative sum

Step-by-step calculation

What the Arithmetic & Geometric Sequence Sums does

What's worth knowing first

Pro tips most people learn the hard way

?What is the Arithmetic & Geometric Sequence Sums?

The Formula

Practical Examples

Frequently Asked Questions

Popular Conversions

Embed this calculator on your site

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Premium Converters

Arithmetic & Geometric Sequence Sums

Arithmetic & Geometric Sequence Sums

Individual terms

Cumulative sum

Step-by-step calculation

What the Arithmetic & Geometric Sequence Sums does

What's worth knowing first

Pro tips most people learn the hard way

?What is the Arithmetic & Geometric Sequence Sums?

The Formula

Practical Examples

Frequently Asked Questions

Arithmetic vs geometric — what's the difference?

When does the infinite geometric sum converge?

Why is Gauss's trick so famous?

Can I have a negative common ratio?

What about non-arithmetic, non-geometric sequences?

Popular Conversions

Embed this calculator on your site

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Premium Converters

Arithmetic & Geometric Sequence Sums

Arithmetic & Geometric Sequence Sums

Individual terms

Cumulative sum

Step-by-step calculation

What the Arithmetic & Geometric Sequence Sums does

What's worth knowing first

Pro tips most people learn the hard way

?What is the Arithmetic & Geometric Sequence Sums?

The Formula

Practical Examples

Frequently Asked Questions

Arithmetic vs geometric — what's the difference?

When does the infinite geometric sum converge?

Why is Gauss's trick so famous?

Can I have a negative common ratio?

What about non-arithmetic, non-geometric sequences?

Popular Conversions

Embed this calculator on your site

More Math Tools

Browse other categories