What's the difference between arithmetic and geometric sequences?

An arithmetic sequence has a constant difference between consecutive terms (e.g., 2, 5, 8, 11 with d=3). A geometric sequence has a constant ratio between consecutive terms (e.g., 2, 6, 18, 54 with r=3). Arithmetic sequences grow linearly while geometric sequences grow exponentially.

When does a geometric series converge?

A geometric series converges (has a finite sum) when the absolute value of the common ratio is less than 1, i.e., |r| = 1, the series diverges.

How do you find the sum of an arithmetic series?

The sum of the first n terms of an arithmetic series is Sn = n(a1 + an)/2, where a1 is the first term and an is the nth term. Equivalently, Sn = n(2a1 + (n-1)d)/2 where d is the common difference. This formula works because you can pair the first and last terms, second and second-to-last, etc.

Sequences & Series Visualizer

Compare arithmetic and geometric sequences. Watch terms grow as bars and track partial sums and convergence behavior.

Sequence Parameters

Presets

First term a₁ = 2.0

2.0

Common difference d = 3.0

3.0

Common ratio r = 2.0

2.0

Number of terms n = 10

Show partial sum

Properties

Type	--
Formula	--
nth term	--
Sum Sₙ	--
Convergence	--
First terms	--

The Math Behind It

Arithmetic Sequences

nth term: aₙ = a₁ + (n-1)d
Sum: Sₙ = n(a₁ + aₙ) / 2
Constant common difference d between terms
Always diverges unless d = 0
Graph of terms is a straight line

Geometric Sequences

nth term: aₙ = a₁ · r^n-1
Sum: Sₙ = a₁(1 - rⁿ) / (1 - r), r ≠ 1
Constant common ratio r between terms
Converges if |r| < 1: S∞ = a₁ / (1 - r)
Diverges if |r| ≥ 1
|r| < 1: terms shrink toward zero

Sequences & Series Visualizer

Sequence Parameters

Properties

The Math Behind It

Arithmetic Sequences

Geometric Sequences

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