Calculate terms and sums for arithmetic and geometric sequences using standard formulas.
Last updated: March 2026 | By Summacalculator
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A sequence is an ordered list of numbers following a specific pattern. Arithmetic sequences have a constant difference between consecutive terms (e.g., 2, 5, 8, 11...), while geometric sequences have a constant ratio (e.g., 3, 6, 12, 24...).
Understanding sequences is fundamental in mathematics, appearing in finance (compound interest), computer science (algorithms), and physics (wave patterns). The formulas allow us to find any term without listing all previous terms, and to calculate the sum of many terms efficiently.
For arithmetic sequences, each term increases by adding the common difference d. For geometric sequences, each term is multiplied by the common ratio r. These simple rules generate infinitely long patterns that model countless real-world phenomena.
aₙ = a₁ + (n-1)d
Sₙ = (n/2)(a₁ + aₙ)
Where d is the common difference
aₙ = a₁ × r^(n-1)
Sₙ = a₁(1-r^n)/(1-r)
Where r is the common ratio (r ≠ 1)
You save $5 in week 1, then increase savings by $3 each week. How much do you save in week 10, and what's the total saved?
By week 10, you'll save $32 that week, with a total of $185 saved over all 10 weeks.
Arithmetic sequences add a constant (e.g., 2, 5, 8, 11...), while geometric sequences multiply by a constant (e.g., 2, 6, 18, 54...). Arithmetic grows linearly, geometric grows exponentially.
Yes! A negative difference creates a decreasing arithmetic sequence (5, 2, -1, -4...). A negative or fractional ratio creates alternating signs or decay in geometric sequences.
When r = 1, all terms are equal to a₁ (constant sequence). The sum formula Sₙ = a₁(1-r^n)/(1-r) fails (division by zero), so we use Sₙ = n×a₁ instead.
For arithmetic: d = a₂ - a₁ (subtract consecutive terms). For geometric: r = a₂ / a₁ (divide consecutive terms). Make sure you have at least two consecutive terms.
When |r| < 1 and n→∞, geometric sequences converge to S = a₁/(1-r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 2. This is fundamental to calculus and fractals.
Yes! That's the power of the formulas. The nth term formula lets you jump directly to any position. For example, find the 100th term instantly without computing the first 99.
Arithmetic: loan amortization, depreciation. Geometric: compound interest, population growth, radioactive decay, computer algorithms (binary search), fractals, and signal processing.
Check consecutive terms. If differences are constant (8-5=3, 11-8=3), use arithmetic. If ratios are constant (6/3=2, 12/6=2), use geometric. If neither, it may be a different type of sequence.
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