Floyd’s Triangle Sum Calculator
Generate Floyd's Triangle up to 500 rows and instantly get the sum of every row plus the total sum of all numbers, using the closed-form row sum formula n(n² + 1)/2.
Floyd's Triangle Generator
Floyd’s Triangle Sum Formula
Where:
- \(n\) — the row number (the row contains exactly \(n\) consecutive integers)
- \(T_n = \frac{n(n+1)}{2}\) — the \(n\)-th triangular number, i.e. the last value in row \(n\)
The first formula gives the sum of the \(n\) numbers in row \(n\) directly, without building the triangle. The second gives the sum of all numbers from row 1 through row \(n\), which is simply the sum of the integers from 1 to \(T_n\).
Floyd’s Triangle Sum – Calculation Example
Example: you want to see the first 5 rows of Floyd's Triangle with the sum of each row and the grand total. Enter Number of rows (n) = 5.
The triangle is:
\[ \begin{array}{ccccc} 1 \\ 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 \end{array} \]
Sum of row 5 using the formula:
\[ \frac{n(n^2 + 1)}{2} = \frac{5(5^2 + 1)}{2} = \frac{5 \times 26}{2} = 65 \]
The last value in row 5 is \(T_5 = \frac{5 \times 6}{2} = 15\), so the total sum of all 15 numbers is:
\[ \frac{15 \times 16}{2} = 120 \]
Result: row sums are 1, 5, 15, 34, 65; the sum of row 5 is 65 and the total sum of all values through row 5 is 120 — exactly \(1 + 2 + \dots + 15\).
How to Use the Calculator
- Enter the desired number of rows in the Number of rows (n) field (any whole number from 1 to 500; the default is 5).
- Click the calculate button.
- Read the generated triangle under Floyd's Triangle: each row is printed on its own line.
- Check the table of row sums — for every row \(n\) you will see its total, which always equals \(\frac{n(n^2+1)}{2}\).
- Find the grand total at the bottom, labeled Total sum of all values: — the sum of every number printed in the triangle.
What Is Floyd's Triangle?
Floyd's Triangle is a triangular array of the natural numbers, named after Robert Floyd. You write the numbers 1, 2, 3, … consecutively, putting one number in the first row, two in the second, three in the third, and so on. Row \(n\) therefore always contains exactly \(n\) numbers, ending with the \(n\)-th triangular number \(T_n = \frac{n(n+1)}{2}\).
The triangle is a classic exercise in introductory programming and number theory because it combines two simple patterns: the counting numbers and the triangular shape of their arrangement. Although the numbers themselves look unremarkable, their row sums behave very regularly. The sum of row \(n\) is \(\frac{n(n^2+1)}{2}\), so the sequence of row sums runs 1, 5, 15, 34, 65, 111, 175, … — a pattern you can verify row by row with this calculator.
The total sum of the whole triangle through row \(n\) is equally tidy. Since the triangle contains every integer from 1 to \(T_n\) exactly once, the total is just \(\frac{T_n(T_n+1)}{2}\). For \(n = 5\) that is \(\frac{15 \times 16}{2} = 120\). This makes it easy to check large triangles without adding hundreds of numbers by hand: generate 100 rows and the total is \(\frac{5050 \times 5051}{2} = 12{,}754{,}275\).
If you enjoy exploring number arrays like this, compare Floyd's Triangle with the Leibniz Harmonic Triangle or a Magic Square generator, two other classic arrangements with surprising sum properties.
Interpreting the Results
The row sum tells you how the "weight" of the triangle grows as you move downward. Because the row sum formula is roughly \(\frac{n^3}{2}\), the totals grow cubically, not linearly — row 10 already sums to 505, and row 50 sums to 62,625. If a computed row sum in your own work differs from \(\frac{n(n^2+1)}{2}\), you know immediately that a number was skipped or duplicated when the triangle was built.
The total sum doubles as a quick identity check: it must always equal a triangular number of a triangular number. It also connects to well-known sums of sequences; if you need those sums on their own, try the Sum of Consecutive Squares or the Arithmetic Progression calculators.
Assumptions and Limitations
- Whole rows only: the input must be a whole number between 1 and 500; the calculator rejects anything outside that range. The upper limit keeps the output readable — 500 rows already print 125,250 numbers.
- Starts at 1: the classic triangle begins with 1 in the first row. Variants that start at another number or use a different step are not covered by the formula shown here.
- Exact integers: all sums are exact integers, so there are no rounding issues; the formulas always produce whole numbers because \(n(n^2+1)\) is even for every \(n\).
- Display size: for large \(n\), the triangle display becomes very long. If you only need the sum of row \(n\) or the total, you can jump straight to the summary table and the total line at the bottom without reading every row.
Frequently Asked Questions
How do I calculate the sum of a row in Floyd's Triangle by hand?
Take the row number n and plug it into the formula \(\frac{n(n^2+1)}{2}\). For row 7: \(\frac{7(49+1)}{2} = \frac{7 \times 50}{2} = 175\). You can verify it by adding the row's numbers directly: row 7 contains 22 through 28, and \(22 + 23 + \dots + 28 = 175\). The formula works because row n is an arithmetic run of n consecutive integers.
Why is the sum of row n equal to n(n² + 1)/2?
Row n starts right after the previous \(T_{n-1} = \frac{(n-1)n}{2}\) numbers and ends at \(T_n\). Its sum is the sum of an arithmetic sequence: the average term \(\frac{T_{n-1}+1+T_n}{2}\) times n terms. Working through the algebra, everything collapses to \(\frac{n(n^2+1)}{2}\). This also proves the result is always an integer, since \(n(n^2+1)\) is even for every n.
What is the sum of all numbers in Floyd's Triangle with n rows?
A triangle with n rows contains every integer from 1 to the n-th triangular number \(T_n = \frac{n(n+1)}{2}\) exactly once, with no gaps or repeats. So the total is just \(\frac{T_n(T_n+1)}{2}\). For 10 rows, \(T_{10} = 55\), giving \(\frac{55 \times 56}{2} = 1{,}540\). This shortcut means you never have to add the rows individually.
Do the row sums grow linearly, quadratically, or cubically?
They grow cubically. Since the row sum is \(\frac{n(n^2+1)}{2} \approx \frac{n^3}{2}\), doubling the row number roughly multiplies the sum by eight. Row 10 sums to 505, but row 20 sums to 4,010 and row 100 sums to 500,050. Keep this in mind when estimating: a 500-row triangle has a total sum in the billions, not millions.
How is Floyd's Triangle different from Pascal's Triangle?
Floyd's Triangle is simply the counting numbers 1, 2, 3, … written in a triangular shape — row n holds n consecutive integers. Pascal's Triangle is built from binomial coefficients, where each entry is the sum of the two above it. They share the triangular layout but have entirely different numbers, sum patterns, and uses. Floyd's Triangle is mostly a programming and number-theory exercise; Pascal's Triangle underpins combinatorics and probability.
Can Floyd's Triangle start at a number other than 1?
Yes, variants exist, but the formulas on this page only apply to the classic version starting at 1. If you start at k instead, row n still has n consecutive integers but the row sum shifts; you would compute it from the first and last values of the row directly using the arithmetic-series average. Similarly, triangles with a step other than 1 (like odd numbers only) need adjusted formulas.
What are common mistakes when computing Floyd's Triangle sums?
The most frequent errors are using n instead of the row length when applying formulas, miscounting the last value of a row (it must equal \(T_n\)), and applying the row formula \(\frac{n(n^2+1)}{2}\) to the total sum. The total instead uses \(\frac{T_n(T_n+1)}{2}\). Also, if your hand-built triangle's row sum doesn't match the closed-form value, a number was skipped or duplicated during generation.