arithmetic progression calculator

What is an arithmetic progression?

An arithmetic progression (AP), also called an arithmetic sequence, is a list of numbers where each term increases or decreases by the same fixed value. That fixed value is called the common difference and is usually written as d.

Examples:

• 2, 5, 8, 11, 14, ... (common difference = +3)
• 40, 35, 30, 25, ... (common difference = -5)
• 7, 7, 7, 7, ... (common difference = 0)

Core formulas used in this AP calculator

• nth term: aₙ = a₁ + (n - 1)d
• Sum of first n terms: Sₙ = n/2 × [2a₁ + (n - 1)d]
• Sum using first and last term: Sₙ = n/2 × (a₁ + aₙ)
• Common difference from first two terms: d = a₂ - a₁
• Number of terms: n = ((aₙ - a₁) / d) + 1

How to use this calculator

1) Find nth term

Select Find nth term, then enter first term, common difference, and n. The tool returns aₙ and a short sequence preview.

2) Find sum of first n terms

Select Find sum of first n terms. Enter a₁, d, and n. You’ll get Sₙ and the nth term used in the sum calculation.

3) Find common difference

Select Find common difference. Enter a₁ and a₂. The calculator returns d immediately.

4) Find number of terms

Select Find number of terms. Enter first term, last term, and d. If values are consistent, the calculator returns n and total sum.

5) Generate sequence

Select Generate sequence and provide a₁, d, and n to display terms in order.

Practical applications

Arithmetic progressions appear in many real scenarios:

• Savings plans: increasing monthly contribution by a fixed amount.
• Loan schedules: fixed step changes in repayment blocks.
• Workout plans: adding a constant number of reps each week.
• Manufacturing: linearly increasing production targets.
• Classroom math: foundational concept for algebra and sequences.

Common mistakes to avoid

• Using n as zero or negative when counting terms.
• Mixing up aₙ and Sₙ (single term vs total sum).
• Forgetting that d can be negative.
• Using inconsistent values where n does not come out as a positive integer.

Quick example

Suppose a₁ = 4, d = 3, and n = 8.

• a₈ = 4 + (8 - 1)×3 = 25
• S₈ = 8/2 × [2×4 + (8 - 1)×3] = 116

This is exactly the kind of calculation the tool above performs automatically.