What Is an Arithmetic Progression Calculator?

An arithmetic progression calculator finds the nth term, sum, and sequence of an AP given the first term, common difference, and number of terms. Pre-filled with a=2, d=5, n=10: the 10th term is 47, sum is 245, mean is 24.5. The linear term chart confirms that APs always form a straight line — their defining characteristic.

Key Formulas

nth term: aₙ = a + (n−1)d. Sum of n terms: Sₙ = n/2 × (2a + (n−1)d) = n/2 × (a₁ + aₙ). Mean of AP: (a₁ + aₙ) / 2 = arithmetic mean. The mean of an AP equals the middle term (or average of two middle terms for even n). Common difference: d = aₙ − aₙ₋₁ (constant for any two adjacent terms).

Gauss's Sum Formula

The sum formula Sₙ = n/2 × (a + l) was reportedly discovered by Carl Friedrich Gauss at age 9 when his teacher asked the class to sum 1 to 100. Gauss noticed: 1+100=101, 2+99=101, 3+98=101, ... 50 pairs each summing to 101. So S₁₀₀ = 50 × 101 = 5050. This pairing insight generalizes to any AP: pair the first and last terms, which always sum to a₁ + aₙ.

APs vs Geometric Progressions

An AP adds a constant (d) to each term; a GP multiplies by a constant (r). APs model linear phenomena: salary with fixed raises, simple interest, linear depreciation. GPs model exponential phenomena: compound interest, population growth, radioactive decay. The term chart of an AP is a straight line; a GP is exponential. Real-world data can often be classified by which fits better.

Applications in Finance

Simple interest payments form an AP: if you borrow $10,000 at 5% simple interest, interest = $500/year, and total payments after n years = principal + n × $500. This contrasts with compound interest (geometric sequence). Annuities with equal payments and no growth are AP-based. Straight-line depreciation: asset loses $D per year, so book value after n years = original cost − n × D (an AP with negative d).