Triangular Number
Sum of the first n natural numbers.
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Example
T_4 = 10
How it works
A triangular number gives the total count when you arrange items in a growing triangle: 1 on the first row, 2 on the second, and so on up to n. The formula works because adding the first n natural numbers can be paired from opposite ends, or seen as half of a rectangle with side lengths n and n+1, which gives n(n+1)/2. This is why the result is always an integer even though we divide by 2.
When to use it
Use this formula when you need the total of a consecutive counting sequence from 1 to n. It appears in counting problems, arrangement patterns, and many basic number theory and combinatorics questions.
Worked example, step by step
Find the 7th triangular number. T_n = n(n+1)/2 T_7 = 7(7+1)/2 T_7 = 7 × 8 / 2 T_7 = 56 / 2 T_7 = 28
Common mistakes to avoid
- Forgetting to multiply by n+1, not just n/2
- Using the formula for the nth term of a sequence without recognizing it gives a cumulative total
- Substituting n incorrectly, such as using n = 6 for the 7th triangular number
Did You Know?
The young Gauss reportedly summed 1 to 100 in seconds by pairing terms, discovering triangular numbers.
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