Arithmetic Sequence Calculator
Calculate nth term and sum of arithmetic sequence
- Enter the first term (a) - This is the starting number of your sequence. For example, if your sequence begins with 2, enter 2.
- Enter the common difference (d) - This is the constant amount added between each term. If your sequence goes 2, 5, 8, the common difference is 3.
- Enter the number of terms (n) - Specify how many terms you want to calculate. Enter a positive integer like 10 to see the first 10 terms.
Click Calculate to see the nth term, sum of all terms, and the complete sequence.
An arithmetic sequence is a list of numbers where each term differs from the previous one by a constant amount. This constant is called the common difference.
The first term (denoted as a or a₁) is where your sequence starts. The nth term (aₙ) is any term at position n in the sequence.
For example, in the sequence 2, 5, 8, 11, 14:
- First term (a₁) = 2
- Common difference (d) = 3 (each term increases by 3)
- 5th term (a₅) = 14
Arithmetic sequences appear everywhere - from the evenly spaced rungs on a ladder to monthly savings with fixed deposits.
nth Term Formula
aₙ = a₁ + (n - 1)d
Find any term by knowing the first term, position, and common difference.
Sum of n Terms (Formula 1)
Sₙ = n/2 × (a₁ + aₙ)
Use when you know the first and nth terms.
Sum of n Terms (Formula 2)
Sₙ = n/2 × [2a₁ + (n - 1)d]
Use when you know the first term and common difference.
Common Difference
d = aₙ₊₁ - aₙ
Find the common difference by subtracting any term from the next term.
| Sequence | Common Difference (d) | Notes |
|---|---|---|
| 2, 5, 8, 11, 14... | 3 | Increasing sequence |
| 10, 7, 4, 1, -2... | -3 | Decreasing sequence |
| 1, 3, 5, 7, 9... | 2 | Odd numbers |
| 5, 5, 5, 5... | 0 | Constant sequence |
Staircase Design
Building codes require uniform riser heights. If each step rises 7 inches, the total height follows an arithmetic sequence: 7, 14, 21, 28 inches...
Seating Arrangements
Theater rows often increase by a fixed number of seats. Row 1 has 20 seats, row 2 has 22, row 3 has 24 - an arithmetic pattern with d = 2.
Salary Increments
Annual raises of a fixed amount create arithmetic growth. Starting at $50,000 with $2,000 yearly raises: $50k, $52k, $54k, $56k...
Depreciation Schedules
Straight-line depreciation reduces asset value by equal amounts each year. A $10,000 asset depreciating $1,000 annually: $10k, $9k, $8k, $7k...
What is an arithmetic sequence?
An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This difference is called the common difference.
How do you find the nth term?
Use the formula aₙ = a₁ + (n - 1)d, where a₁ is the first term, d is the common difference, and n is the position of the term you want to find.
What is the formula for the sum?
The sum of the first n terms is Sₙ = n/2 × (a₁ + aₙ) or Sₙ = n/2 × [2a₁ + (n - 1)d]. Both formulas give the same result.
Can the common difference be negative?
Yes. A negative common difference creates a decreasing sequence. For example, 10, 7, 4, 1 has d = -3.
How is arithmetic sequence different from geometric?
In arithmetic sequences, you add a constant difference. In geometric sequences, you multiply by a constant ratio. Arithmetic: 2, 5, 8, 11 (add 3). Geometric: 2, 6, 18, 54 (multiply by 3).
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