Sum Of Linear Number Sequence
Sum of first n natural numbers using formula n(n+1)/2. Free online tool with step-by-step formula breakdown for sequence 1+2+3+...+n with instant results.
About This Calculator
The sum of the first n natural numbers (also called the sum of a linear number sequence) is one of the most fundamental results in arithmetic. The formula S = n(n+1)/2 gives the total of 1+2+3+...+n for any positive integer n. This classic result has been known since ancient times and is one of the first mathematical series students learn.
This sum of linear number sequence calculator applies the well-known formula S = n(n+1)/2 to compute the sum instantly. For example, entering n = 100 gives 100x101/2 = 5050, which is the famous sum that the mathematician Carl Friedrich Gauss computed in seconds as a schoolchild. The calculator also shows the step-by-step formula derivation so you can see exactly how each result is obtained.
The resulting sum of the first n natural numbers is called the nth triangular number, visualized as a triangular arrangement of dots. Triangular numbers appear throughout mathematics, from the handshake problem (a group of n people shakes n(n-1)/2 hands) to the binomial coefficient C(n+1, 2) used in combinatorics and probability.
Regional Notes
The sum of natural numbers formula is universally taught in mathematics curricula worldwide. In India, it is covered in CBSE Class 10 arithmetic progressions chapters. In the US, students learn it in middle school and high school algebra. In the UK, it appears in GCSE mathematics as part of sequences and series. The formula and proof are identical across all educational systems.
Frequently Asked Questions
What is the sum of first n natural numbers formula?
The sum of the first n natural numbers is given by the formula S = n(n+1)/2. For example, the sum of 1+2+3+...+100 = 100x101/2 = 5050. This formula was discovered by the mathematician Carl Friedrich Gauss and is also known as the triangular number formula.
How was the sum of natural numbers formula derived?
The formula can be derived by pairing terms: 1 + 2 + 3 + ... + n = (1+n) + (2+(n-1)) + (3+(n-2)) + ... = (n+1) x n/2. Each pair sums to n+1, and there are n/2 such pairs. This elegant derivation is often attributed to Carl Friedrich Gauss who discovered it as a schoolchild.
What are triangular numbers?
Triangular numbers are the sequence of numbers that represent the sum of the first n natural numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and so on. Each triangular number Tₙ = n(n+1)/2 can be visualized as a triangular arrangement of dots where the base has n dots.
What are the real-world applications of the sum of natural numbers?
The sum of natural numbers appears in combinatorics (counting combinations), computer science (algorithm analysis, comparing N^2/2 operations), statistics (pairwise comparisons, rank-sum tests), finance (summing arithmetic progressions), and geometry (triangular arrangements in construction and design).
How is the sum of linear number sequence related to the arithmetic series?
The sum 1+2+3+...+n is a special case of an arithmetic series with first term a=1 and common difference d=1. Plugging into the arithmetic series formula S = n/2 x (2a + (n-1)d) gives S = n/2 x (2 + n - 1) = n(n+1)/2, confirming the same result.
Is this sum of natural numbers calculator free?
Yes, all calculators on Calculy are completely free to use. Enter any positive integer n to instantly get the sum of 1+2+3+...+n with the formula breakdown showing n(n+1)/2 step by step. No registration or payment is required.