arithmetic sequence calculator

If you have ever added a fixed amount every week, increased your workout reps by the same number each session, or tracked a steady monthly change in revenue, you have already worked with an arithmetic sequence. This arithmetic sequence calculator helps you quickly compute terms and sums without doing repetitive manual math.

What Is an Arithmetic Sequence?

An arithmetic sequence (also called an arithmetic progression) is a list of numbers where the difference between consecutive terms is constant. That constant is called the common difference.

• Example with positive difference: 5, 8, 11, 14, ... (difference = +3)
• Example with negative difference: 20, 16, 12, 8, ... (difference = -4)
• Example with zero difference: 7, 7, 7, 7, ... (difference = 0)

Core Formulas Used in This Calculator

1) Formula for the k-th term

a_k = a₁ + (k - 1)d

Where:

• a₁ = first term
• d = common difference
• k = position of the term you want

2) Formula for the sum of the first n terms

S_n = n / 2 × [2a₁ + (n - 1)d]

This gives you the total of the first n values in the sequence, which is useful for budgeting, savings schedules, and forecasting.

How to Use This Arithmetic Sequence Calculator

1. Enter the first term.
2. Enter the common difference.
3. Enter how many terms you want to include (n).
4. Optionally enter a specific term position (k) to find.
5. Click Calculate.

You’ll get:

• The value of the k-th term
• The sum of the first n terms
• A preview of the generated sequence

Worked Examples

Example 1: Weekly Savings Plan

You save $10 in week 1 and increase by $5 each week.

• a₁ = 10
• d = 5
• n = 12 weeks

Using the formula, the 12th week savings amount is 65, and the total saved over 12 weeks can be found with the sum formula.

Example 2: Stair-Step Training

Your push-up plan starts at 15 reps and increases by 2 reps per day.

• a₁ = 15
• d = 2
• k = 20

The 20th day target is a₂₀ = 53 reps.

Common Mistakes to Avoid

• Confusing arithmetic and geometric sequences: arithmetic uses addition/subtraction by a fixed amount; geometric uses multiplication/division by a fixed ratio.
• Using the wrong index: remember the first term is position 1, not 0.
• Sign errors: a negative common difference means the sequence decreases.
• Mixing n and k: n is typically used for count/sum; k is a specific term index.

When This Tool Is Useful

This calculator is practical for:

• Personal finance planning (incremental deposits or payments)
• Classroom math checks and homework verification
• Habit tracking with linear changes
• Simple business forecasting with steady increments

Quick FAQ

Can the common difference be negative?

Yes. A negative difference creates a decreasing arithmetic sequence.

Can I use decimal values?

Absolutely. The calculator accepts decimals for both the first term and common difference.

Is this the same as a geometric sequence calculator?

No. Geometric sequences are based on multiplication by a common ratio, not addition by a common difference.

Final Thoughts

An arithmetic sequence calculator saves time and reduces mistakes, especially when you need both individual terms and total sums quickly. Use it as a learning companion, a planning tool, or a fast way to validate your own hand calculations.