Sum of Arithmetic Series Calculator
Calculate the sum of an arithmetic series with our free online calculator. Enter the first term, common difference, and number of terms to find the sum instantly with step-by-step solutions.
Understanding Arithmetic Series
An arithmetic series is the sum of an arithmetic sequence – a list of numbers where each term increases or decreases by the same amount. That constant amount is called the "common difference." You see arithmetic series everywhere: calculating total savings with regular deposits, finding the sum of consecutive numbers, or working out cumulative distances when speed changes steadily.
The beauty of arithmetic series is that you don't need to add every single term. There's a clean formula that gives you the sum instantly, no matter how many terms you have.
The Sum Formula
Sₙ = n/2 × (a₁ + aₙ)
Where Sₙ = sum, n = number of terms, a₁ = first term, aₙ = last term
If you don't know the last term, you can find it first using: aₙ = a₁ + (n-1)d, where d is the common difference. Then plug it into the sum formula. Some people prefer the combined version: Sₙ = n/2 × [2a₁ + (n-1)d], which skips the intermediate step.
Worked Examples
Example 1: Sum of the first 15 terms starting at 5 with difference 3
Sequence: 5, 8, 11, 14, 17, ... (15 terms total)
Step 1: Find the 15th term
a₁₅ = 5 + (15-1) × 3 = 5 + 42 = 47
Step 2: Apply the sum formula
S₁₅ = 15/2 × (5 + 47) = 7.5 × 52 = 390
The sum of all 15 terms is 390.
Example 2: Sum of integers from 1 to 100
This is the famous problem young Gauss supposedly solved in seconds.
a₁ = 1, d = 1, n = 100
a₁₀₀ = 1 + (100-1) × 1 = 100
S₁₀₀ = 100/2 × (1 + 100) = 50 × 101 = 5,050
The sum of all integers from 1 to 100 is 5,050.
Example 3: Decreasing sequence with negative difference
Find the sum of 8 terms starting at 10 with common difference -2.
Sequence: 10, 8, 6, 4, 2, 0, -2, -4
a₈ = 10 + (8-1) × (-2) = 10 - 14 = -4
S₈ = 8/2 × (10 + (-4)) = 4 × 6 = 24
Even with negative terms, the formula works the same way. The sum is 24.
Quick Fact
The story goes that 8-year-old Carl Friedrich Gauss was told to add all numbers from 1 to 100 as busywork. He instantly realized that pairing 1+100, 2+99, 3+98, and so on always gives 101, and there are 50 such pairs. So 50 × 101 = 5,050. This insight is essentially the arithmetic series formula in disguise. Whether the story is true or not, it's a brilliant way to understand why the formula works.
Frequently Asked Questions
What's the difference between an arithmetic sequence and an arithmetic series?
A sequence is the list of numbers itself (like 2, 5, 8, 11, 14). A series is the sum of those numbers (2 + 5 + 8 + 11 + 14 = 40). The sequence shows the pattern; the series gives you the total.
Can the common difference be negative?
Absolutely. A negative common difference means the sequence decreases. For example, 20, 17, 14, 11, 8 has a common difference of -3. The sum formula works exactly the same way – just plug in the negative value for d.
What if I only know the first term and common difference, not the last term?
No problem. First calculate the last term using aₙ = a₁ + (n-1)d, then use the sum formula. Or use the combined formula Sₙ = n/2 × [2a₁ + (n-1)d] which does both steps at once.
Does this work for decimal or fractional terms?
Yes. The formula doesn't care whether your terms are integers, decimals, or fractions. As long as the difference between consecutive terms is constant, it's an arithmetic series and the formula applies.
How is this different from a geometric series?
In an arithmetic series, you add the same amount each time (like +3). In a geometric series, you multiply by the same amount each time (like ×2). They have completely different formulas. Arithmetic series grow linearly; geometric series grow exponentially.
Why does the formula use n/2? Where does that come from?
Think of pairing terms from opposite ends: first + last, second + second-to-last, and so on. Each pair sums to the same value (a₁ + aₙ). With n terms, you have n/2 such pairs. That's why the formula is n/2 × (a₁ + aₙ). It's the same insight Gauss supposedly had as a child.
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