The sequence of triangular numbers is the sequence of numbers $t_{0}, t_{1}, t_{2}, \ldots$ defined by $t_{0}=0$ and for any index $k, t_{k+1}=t_{k}+k+1$.

![](https://cdn.mathpix.com/cropped/2024_05_10_169e018874c7653739feg-36.jpg?height=461&width=483&top_left_y=1645&top_left_x=840)

We see from the drawing that $t_{5}=t_{4}+5=10+5=15$.

Determine an expression for $t_{n}$ in terms of $n$.
We look at small values: $t_{0}=0, t_{1}=1, t_{2}=3, t_{3}=6, t_{4}=10$, $t_{5}=15$. We then intuit that $t_{n}=n(n+1) / 2$.

$\triangleright$ The formula is verified for $n=0$.

$\triangleright$ Let $n$ be an integer such that $t_{n}=n(n+1) / 2$. Then,

$$
t_{n+1}=t_{n}+n+1=\frac{(n+1)(n+2)}{2}
$$

We conclude by induction.

## Remark

It is said that as a child, Gauss used the following method to calculate $t_{100}$

$$
\begin{aligned}
& t_{100}=1+2+\ldots+99+100
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_10_169e018874c7653739feg-37.jpg?height=111&width=921&top_left_y=622&top_left_x=491)

$\mathrm{d}^{\prime}$ where $t_{100}=100 \times 101 / 2$. Remark

We can similarly define square numbers (the $n$-th is $n^{2}$):

![](https://cdn.mathpix.com/cropped/2024_05_10_169e018874c7653739feg-37.jpg?height=368&width=474&top_left_y=1060&top_left_x=717)

and pentagonal numbers (the $n$-th is $n(3 n-1) / 2$):

![](https://cdn.mathpix.com/cropped/2024_05_10_169e018874c7653739feg-37.jpg?height=688&width=705&top_left_y=1594&top_left_x=583)