What is 1 + 2 + ⋯ (till n)?

Saturday, June 04, 2022

The answer is:

n × (n + 1) / 2

Let's check with an example, n = 4.

1 + 2 + 3 + 4
= 4 × (4 + 1) / 2
= 4 × 5 / 2
= 10

Yep. That's right.

Let's see if we can prove this...

(Continue scrolling to play the animation. Please use a smooth scrolling device like a touchscreen or a trackpad.)

Let's represent 1 to n with balls.
The number of yellow balls =
1 + 2 + 3 + ⋯ (till n)
Double the number of balls,
and make a grid.
The first row has 1 yellow & n blue balls.
A total of n + 1 balls.
The second row has 2 yellow & n-1 blue balls.
Again, a total of n + 1 balls.
Every row has n + 1 balls!
The grid has a total of n × (n + 1) balls.
The number of yellow balls
is half that = n × (n + 1) / 2
Therefore,
1 + 2 + ⋯ (till n)  =  n × (n + 1) / 2

(@-me on twitter @kssreeram for comments and feedback.)