Closed Form for Triangular Numbers/Also presented as
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Closed Form for Triangular Numbers: Also presented as
The closed-form expression for the $n$th triangular number can also be presented as:
- $\ds T_n = \sum_{i \mathop = 0}^n i = \frac {n \paren {n + 1} } 2$
This is seen to be equivalent to the given form by the fact that the first term $T_0$ evaluates as $\dfrac {0 \paren {0 + 1} } 2$ which is zero.
Sources
- 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: Exercises: $1.2 \ \text {a)}$
- 1982: P.M. Cohn: Algebra: Volume $\text { 1 }$ (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients