Sum Of Series
Calculate the sum of arithmetic and geometric series up to n terms. Enter first and second terms plus number of terms for instant sum results with formula identification.
About This Calculator
A series is the sum of the terms of a sequence. Understanding how to compute series sums is essential in mathematics, physics, finance, and engineering. The two most fundamental types are arithmetic series (sum of terms with constant difference) and geometric series (sum of terms with constant ratio).
This sum of series calculator handles both types automatically. For arithmetic series, it computes the common difference d from the first two terms (d = second - first) and applies Sₙ = n/2 x (2a + (n-1)d). For geometric series, it computes the common ratio r (r = second/first) and applies Sₙ = a(1-rⁿ)/(1-r) when r ≠ 1. The calculator correctly handles edge cases including r = 1 and negative values for d or r.
The results display the total sum (rounded to 2 decimal places), the identified series type (arithmetic or geometric), and the number of terms used in the calculation. This tool is valuable for students preparing for exams, financial analysts computing investment returns, and engineers analyzing periodic patterns in data.
Regional Notes
Series sums are a universal mathematical concept taught worldwide. In India, arithmetic and geometric series are covered extensively in CBSE Class 11 and Class 12 mathematics, including applications in JEE preparation. In the US, series appear in Algebra 2, Precalculus, and AP Calculus courses. In the UK, series are part of A-Level mathematics, including the geometric series convergence condition |r| < 1.
Frequently Asked Questions
What is the difference between an arithmetic series and a geometric series?
An arithmetic series sums the terms of an arithmetic sequence where consecutive terms have a constant difference (e.g., 3+7+11+15). A geometric series sums the terms of a geometric sequence where consecutive terms have a constant ratio (e.g., 3+6+12+24). This calculator handles both types using their respective formulas.
What is the formula for the sum of an arithmetic series?
The sum of the first n terms of an arithmetic series is Sₙ = n/2 x (2a + (n-1)d), where a is the first term and d is the common difference. Given the first term a and second term, the common difference d = second - first. The formula derives from pairing terms that sum to the same value.
What is the formula for the sum of a geometric series?
The sum of the first n terms of a geometric series is Sₙ = a(1-rⁿ)/(1-r) when r ≠ 1, where a is the first term and r is the common ratio (r = second term / first term). When r = 1, all terms equal a, so Sₙ = a x n. This calculator applies the correct formula automatically.
What are real-world applications of series sums?
Arithmetic series sums model total savings with regular deposits, accumulated rental income, or total distance with constant acceleration. Geometric series sums model compound interest calculations, loan amortization, depreciation schedules, multi-year investment returns, and total revenue with constant growth rates.
Can the sum of an infinite geometric series be calculated?
Yes, an infinite geometric series converges to a finite sum S = a/(1-r) when the absolute value of the common ratio |r| < 1. For example, 1 + 1/2 + 1/4 + 1/8 + ... = 2. When |r| >= 1, the infinite series diverges and has no finite sum. This calculator computes finite sums up to n terms.
How do I identify whether a series is arithmetic or geometric?
Check the relationship between consecutive terms. If the difference between consecutive terms is constant (e.g., 5, 8, 11, 14 has difference 3), it is arithmetic. If the ratio between consecutive terms is constant (e.g., 5, 10, 20, 40 has ratio 2), it is geometric. If neither, it may be another type of sequence.
Is this series calculator free to use?
Yes, all calculators on Calculy are completely free to use. There are no subscription fees, usage limits, or hidden charges. You can calculate as many arithmetic and geometric series sums as you need, and the results include the sum value and series type identification.