Sum of Series Calculator
Calculate sum of arithmetic and geometric series
Instructions
Select Series Type
Choose between arithmetic series or geometric series.
Enter Values
Input the first term, common difference/ratio, and number of terms to sum.
Calculate Sum
Click "Calculate Sum" to see the sum of the series displayed.
Formulas
Arithmetic: S_n = n/2 × [2a₁ + (n-1)d]
where d is the common difference
Geometric: S_n = a₁ × (1 - rⁿ) / (1 - r), for r ≠ 1
If r = 1, then S_n = a₁ × n
Example 1: Sum of arithmetic series 2 + 5 + 8 + 11 + 14
a₁ = 2, d = 3, n = 5
S₅ = 5/2 × [2×2 + (5-1)×3] = 5/2 × [4 + 12] = 5/2 × 16 = 40
Example 2: Sum of geometric series 2 + 6 + 18 + 54
a₁ = 2, r = 3, n = 4
S₄ = 2 × (1 - 3⁴) / (1 - 3) = 2 × (1 - 81) / (-2) = 2 × (-80) / (-2) = 80
About Sum of Series Calculator
The sum of series calculator helps you find the sum of arithmetic and geometric series. A series is the sum of terms in a sequence, and this calculator provides quick calculations for both common types.
Arithmetic Series
- Sum of terms in an arithmetic sequence
- Example: 2 + 5 + 8 + 11 + 14 = 40
- Used in calculations involving linear patterns
Geometric Series
- Sum of terms in a geometric sequence
- Example: 2 + 6 + 18 + 54 = 80
- Used in exponential growth calculations, financial modeling
Common Questions
What's the difference between a sequence and a series?
A sequence is a list of numbers (2, 5, 8, 11), while a series is the sum of those numbers (2 + 5 + 8 + 11 = 26).
Can I find the infinite sum of a geometric series?
For infinite geometric series, if |r| < 1, the sum converges to S_∞ = a₁ / (1 - r). This calculator focuses on finite sums.
What if the common ratio equals 1?
If r = 1 in a geometric series, all terms are equal, and the sum is simply S_n = a₁ × n.
Can I use negative values?
Yes! Both arithmetic and geometric series work with negative first terms and negative common differences/ratios.
How accurate are the results?
The calculator provides high precision results. Results are displayed with up to 6 decimal places for clarity.