Arithmetic Sequence Calculator
This calculator helps you work with arithmetic sequences (also called arithmetic progressions). You can find the nth term or the sum of the first n terms.
Understanding Arithmetic Sequences
An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant difference, d, to the previous term.
- nth term formula: aₙ = a₁ + (n-1) × d
- Sum of first n terms: Sₙ = n/2 × (2a₁ + (n-1)d)
Example: Sequence: 2, 5, 8, 11,… → a₁ = 2, d = 3 → 5th term: 2 + (5-1)×3 = 14
How the Calculator Works
The calculator automatically:
Step 1: Takes input values: a₁, d, n
Step 2: Selects calculation type: nth term or sum
Step 3: Applies the arithmetic sequence formula
Step 4: Displays result instantly
Examples
1. a₁ = 3, d = 2, n = 5 → nth term: 3 + (5-1)×2 = 11
2. a₁ = 1, d = 4, n = 6 → sum of first 6 terms: 6/2 × (2×1 + (6-1)×4) = 3 × 22 = 66
3. a₁ = 5, d = -1, n = 8 → nth term: 5 + (8-1)(-1) = -2
Why Arithmetic Sequences Matter
Arithmetic sequences appear in finance (installments, savings), physics, and algebra problem-solving. They help identify patterns and calculate sums quickly.
Common Mistakes to Avoid
- Using wrong formula (nth term vs sum)
- Forgetting to multiply (n-1) with d for nth term
- Miscalculating sum with wrong n value
Important: For nth term, always multiply (n-1) by the common difference before adding to the first term.
Correct: aₙ = a₁ + (n-1)d, not a₁ + n×d.
Frequently Asked Questions
Can d be negative?
Yes. Sequences can increase or decrease depending on the common difference.
Can I use decimals?
Yes. a₁, d, and n can be decimals (for nth term) or integers.
What if n = 1?
The first term a₁ is the nth term; sum = a₁.
Conclusion
This arithmetic sequence calculator is a quick and reliable tool to find the nth term or the sum of first n terms, helping students, professionals, and math enthusiasts solve sequence problems easily.