Question

Examine the diagram on the right and explain how it illustrates the value of n^+n/2

Answer 1

the key is to write
`n^2 + n` as
`n(n+1)` which can be easily seen as an area of a rectangle with one side length n and the other (n+1). That explains the rectangle made of dots - 3 x 4

Now you only have half of them because of the 1/2. So that explains the lower "triangle" of dots being greyed out - they do not count in the area (upper triangle). So the count of the dark dots corresponds to the original expression.

For n=5 you can draw a similar rectangular matrix with 5 x 6 dots. And count only the upper triangular part.