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Arithmetic & Geometric Sequence Sums

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 April 25, 2026

a_10
29
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).

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

?What is the Arithmetic & Geometric Sequence Sums?

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. Used for math homework, financial planning (regular savings = arithmetic; compound growth = geometric), and intuition-building about exponential vs linear growth.

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

1

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

2

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

3

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

4

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

5

Loan amortization mathematically combines arithmetic + geometric series.

6

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). Geometric MULTIPLIES by a constant (r) each step: 2, 6, 18, 54, … (r = 3). Geometric grows much faster — exponentially vs linearly.

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