Sigma Notation and Triangular Numbers

Sigma notation and triangular numbers

$\sum_{k=1}^{n} k = 1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}{2}$

Step-by-step: $\sum_{k=1}^{6} k$

$1 + 2 + 3 + 4 + 5 + 6 = \frac{6 \times 7}{2} = 21$

Why "triangular numbers"?

Arrange dots in a triangle: row 1 has 1 dot, row 2 has 2, ..., row $n$ has $n$. The total is $1 + 2 + \cdots + n = n(n+1)/2$. These are called triangular numbers: 1, 3, 6, 10, 15, 21, 28, ...

The proof (Gauss's trick)

Write $S = 1 + 2 + \cdots + n$ forward and backward, add: $2S = n$ pairs of $(n+1)$. So $S = n(n+1)/2$.

Try it in the visualization

Watch dots form a triangle as $n$ increases. The pairing trick is shown: the triangle doubles into a $n \times (n+1)$ rectangle, so the triangle has half that area. Visualized as a triangle of dots: $n$ rows, with row $k$ containing $k$ dots. The formula counts the total dots.