Triangular Number CalculatorFree online math calculator — instant results, no sign-up
Calculate the sum of integers from one through n.
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About the Triangular Number Calculator
The Triangular Number Calculator adds every integer from 1 through your chosen upper integer, n, including both endpoints. It uses the triangular-number formula to find the total without adding each term separately. With the default upper integer of 100, the result is 5,050.
- Upper integer
- The final whole number n included in the sum from 1 through n; it is dimensionless and defaults to 100.
Formula and method
T_n = 1 + 2 + … + n = n(n + 1) / 2; for n = 100, T_100 = 100 × 101 / 2 = 5,050.
- nThe upper integer and the number of terms in the sum from 1 through n.
- T_nThe nth triangular number: the sum of all integers from 1 through n.
Multiply the upper integer by the next integer, then divide the product by 2. This formula follows from pairing the sequence with its reverse: every pair sums to n + 1, and there are n pairs across the two copies of the sum. For the default input, 100 × 101 = 10,100, and dividing by 2 gives 5,050.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Upper integer100
Results
- Result5,050
How to use the Triangular Number Calculator
- 1Enter the last integer you want included in the Upper integer field.
- 2For the worked example, leave the default value at 100.
- 3Calculate the triangular number using n(n + 1) / 2.
- 4Read the Result row: for 100, it shows 5,050, the total of all integers from 1 through 100.
Reading your result
The Result row displays T_n, the complete sum from 1 through the upper integer, not the number of terms or their average. With Upper integer set to 100, Result = 5,050 means that adding every integer from 1 through 100 produces 5,050. The result is dimensionless.
Limitations
The stated sum assumes a positive integer upper bound and consecutive terms starting at 1 with a step of 1. Fractions and negative values do not represent this finite sum, even though the algebraic expression can be evaluated for them. By convention, n = 0 gives an empty sum of 0, if the tool accepts zero. Very large inputs may exceed the calculator's numeric precision or display limits; no specific maximum is established here.
Common questions
Does the sum include the upper integer?+
Yes. With an upper integer of 100, the calculator includes 100 as the final term, giving a total of 5,050.
Why is the result called a triangular number?+
The total counts objects arranged in triangular rows: the first row has one object, the next has two, and each following row has one more, through row n.
Why does the triangular-number formula divide by 2?+
Pairing the forward and reversed sequences gives n pairs, each totaling n + 1. That counts the original sum twice, so n(n + 1) must be divided by 2.
Can this calculator sum only a selected range that starts above 1?+
Not directly: its single input sets the endpoint of a sum starting at 1. For a range from a through n, subtract the triangular number for a − 1 from the triangular number for n.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.