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CalcivoTools

Arithmetic Sequence Calculator : Find nth Term (a_n) & Arithmetic Series Sum (S_n)

The Calcivo Arithmetic Sequence Calculator solves arithmetic progressions (AP). Given the first term (a₁), common difference (d), and term position (n), it calculates the exact value of the nth term and the cumulative series sum (S_n).

Interactive Calculator

100% Client-Side Engine
Value of nth Term (a20)
62
Sum of First 20 Terms (S20)
670
First 10 Generated Terms
[5, 8, 11, 14, 17, 20, 23, 26, 29, 32, ...]
Arithmetic Progression Derivation
1. nth Term Formula: a_n = a_1 + (n - 1)d
a_20 = 5 + (20 - 1)(3) = 5 + 57 = 62
2. Series Sum Formula: S_n = \frac{n}{2}(a_1 + a_n)
S_20 = (20/2) × (5 + 62) = 670

Step-by-Step Examples

20th Term of Progression Starting at 5 with d = 3

a₁ = 5, d = 3, n = 20.

Inputs: First Term (a₁): 5, Difference (d): 3, Position (n): 20
Result: a₂₀ = 62, S₂₀ = 670

a₂₀ = 5 + (19 × 3) = 62. S₂₀ = (20 / 2) × (5 + 62) = 10 × 67 = 670.

How to Use This Tool

  1. Enter the first term (a₁).
  2. Enter the constant common difference (d).
  3. Enter the target term position (n).
  4. Click Calculate to see the nth term, series sum, and progression table.

Frequently Asked Questions

Pairing terms from both ends (first + last, second + second-to-last) always yields the same sum (a₁ + a_n). Multiplying by n / 2 gives the total sum S_n.

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