# Closed Form for Triangular Numbers/Proof by Induction

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## Theorem

The closed-form expression for the $n$th triangular number is:

- $\displaystyle T_n = \sum_{i \mathop = 1}^n i = \frac {n \paren {n + 1} } 2$

## Proof

Proof by induction:

### Basis for the Induction

When $n = 1$, we have:

- $\ds \sum_{i \mathop = 1}^1 i = 1$

Also:

- $\dfrac {n \paren {n + 1} } 2 = \dfrac {1 \cdot 2} 2 = 1$

This is our base case.

### Induction Hypothesis

- $\forall k \in \N: k \ge 1: \ds \sum_{i \mathop = 1}^k i = \frac {k \paren {k + 1} } 2$

This is our induction hypothesis.

It is to be demonstrated that:

- $\ds \sum_{i \mathop = 1}^{k + 1} i = \frac {\paren {k + 1} \paren {k + 2} } 2$

### Induction Step

This is our induction step:

Consider $n = k + 1$.

By the properties of summation:

- $\ds \sum_{i \mathop = 1}^{k + 1} i = \sum_{i \mathop = 1}^k i + k + 1$

Using the induction hypothesis this can be simplified to:

\(\ds \frac {k \paren {k + 1} } 2 + k + 1\) | \(=\) | \(\ds \frac {k \paren {k + 1} + 2 k + 2} 2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {k^2 + 3 k + 2} 2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\paren {k + 1} \paren {k + 2} } 2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\paren {k + 1} \paren {\paren {k + 1} + 1} } 2\) |

Hence the result by induction.

$\blacksquare$

## Also see

This is usually the first proof by induction that a student mathematician encounters.

The second one is often Proof by Induction of Sum of Sequence of Squares.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {1-1}$ Principle of Mathematical Induction - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 20 \beta$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 3$: Natural Numbers: $\S 3.9$: Example - 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Problems $1.1$: $1 \ \text {(a)}$ - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.4.2$: Mathematical induction: Theorem $1.30$

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