Arithmetic & Geometric Sequence Calculator
Find the nth term of either an arithmetic or geometric sequence side by side. Perfect for students comparing how linear and exponential growth differ over the same number of steps.
About this calculator
An arithmetic sequence adds a fixed common difference d each step, giving the nth term: aₙ = a₁ + (n − 1) × d. A geometric sequence multiplies by a fixed common ratio r each step, giving: aₙ = a₁ × rⁿ⁻¹. Both formulas share the same structure — a starting value modified by n − 1 steps — but arithmetic growth is linear while geometric growth is exponential. The sum of the first n arithmetic terms is Sₙ = (n/2)(2a₁ + (n−1)d), while the sum of n geometric terms is Sₙ = a₁(rⁿ − 1)/(r − 1). Choosing the right model matters: salaries with flat annual raises follow arithmetic sequences, while investment accounts with percentage returns follow geometric ones. This calculator lets you switch between types instantly to compare both behaviors.
How to use
Say a₁ = 5, and you want the 8th term with common difference/ratio = 3. Arithmetic (sequence_type = arithmetic): a₈ = 5 + (8 − 1) × 3 = 5 + 21 = 26. Geometric (sequence_type = geometric): a₈ = 5 × 3⁸⁻¹ = 5 × 3⁷ = 5 × 2187 = 10,935. The contrast is striking — by term 8 the geometric sequence is already over 400 times larger than the arithmetic one, illustrating the power of exponential growth.
Frequently asked questions
What is the difference between arithmetic and geometric sequences?
In an arithmetic sequence, a constant value (the common difference d) is added to each term to get the next one, producing linear growth. In a geometric sequence, each term is multiplied by a constant (the common ratio r), producing exponential growth or decay. For small n the sequences may look similar, but geometric sequences grow or shrink far more rapidly for large n. Arithmetic sequences are used for evenly spaced quantities; geometric sequences model percentage-based change.
How do I determine whether a sequence is arithmetic or geometric?
Check the differences between consecutive terms first. If aₙ₊₁ − aₙ is the same constant for every pair, the sequence is arithmetic. If those differences are not constant, check the ratios: if aₙ₊₁ / aₙ is the same constant for every pair, the sequence is geometric. If neither the differences nor the ratios are constant, the sequence is neither type and requires a different model. Having just three consecutive terms is usually enough to make the determination.
Why does a geometric sequence grow so much faster than an arithmetic sequence with the same starting values?
Arithmetic growth adds the same fixed amount every step, so the total increase after n steps is proportional to n. Geometric growth multiplies by the same factor every step, so after n steps the value is proportional to rⁿ — an exponential function. Exponential functions always outpace linear ones for sufficiently large n, regardless of starting values. This is why compound interest (geometric) creates vastly more wealth over decades than simple interest (arithmetic) at the same nominal rate.