Number Sequence Calculator · Practical guide
Sum of the First 100 Natural Numbers
1 + 2 + 3 + … + 100 = 5050. Here is the 10-second method behind the answer.
At a glance
The Answer
The sum of the first 100 natural numbers is 5050. You can get there without adding a hundred numbers by hand: the sequence is arithmetic with first term a₁ = 1 and common difference d = 1.
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Enter your own values, then compare the result with the examples below.
Arithmetic Sequence Result
First 10 terms:
Geometric Sequence Result
First 10 terms:
The Formula Method
Using the arithmetic series formula Sₙ = n/2 × (2a₁ + (n − 1)d) with n = 100, a₁ = 1, d = 1: S₁₀₀ = 100/2 × (2 + 99) = 50 × 101 = 5050. The shortcut version for consecutive integers starting at 1 is simply Sₙ = n(n + 1)/2 = 100 × 101/2 = 5050.
The Pairing Trick
The same logic works by pairing terms from opposite ends: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on. There are 50 such pairs, each summing to 101, giving 50 × 101 = 5050. Legend says the young Gauss used exactly this trick when his teacher assigned the sum as busywork.
Sums of Other Ranges
| Range | Sum |
|---|---|
| 1 to 10 | 55 |
| 1 to 50 | 1,275 |
| 1 to 100 | 5,050 |
| 1 to 1,000 | 500,500 |
To verify any of these — or to sum a sequence with a different starting value or step, like 5, 9, 13, 17, … — enter the first term, common difference, and term count in the calculator above. It returns both the nth term and the full sum of the first n terms.
Frequently Asked Questions
What is the formula for the sum of the first n natural numbers?
S = n(n + 1)/2. This is a special case of the arithmetic series formula where the first term is 1 and the common difference is 1.
What is the sum of the first 100 even numbers?
The sequence 2, 4, 6, …, 200 has a₁ = 2 and d = 2, so S₁₀₀ = 100/2 × (4 + 99 × 2) = 50 × 202 = 10,100.
What is the sum of the first 100 odd numbers?
The sequence 1, 3, 5, … gives S₁₀₀ = 100/2 × (2 + 99 × 2) = 50 × 200 = 10,000 — which equals 100², since the sum of the first n odd numbers is always n².