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The triangular numbers are defined by the series
That is, the  th triangular number is simply the sum of the first  natural numbers. The first few triangular numbers are
The name triangular number comes from the fact that the summation defining  can be visualized as the number of dots in
where the number of rows is equal to  .
The closed-form for the triangular numbers is
Legend has it that a grammar-school-aged Gauss was told by his teacher to sum up all the numbers from 1 to 100. He reasoned that each number  could be paired up with  , to form a sum of  , and if this was done  times, it would result in twice the actual sum (since each number would get used twice due to the pairing). Hence, the sum would be
The same line of reasoning works to give us the closed form for any  .
Another way to derive the closed form is to assume that the  th triangular number is less than or equal to the  th square (that is, each row is less than or equal to  , so the sum of all rows must be less than or equal to  or  ), and then use the first few triangular numbers to solve the general 2nd degree polynomial
 for  ,  , and  . This leads to  ,  , and  , which is the same as the above formula for  .
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