Consider the interval: $[1,n]$
How many subintervals does this interval have?
For example, $n = 3$ has 6 subintervals: $[1], [2], [3], [1,2], [1,3], [2,3]$
2026-04-13 15:42:37.1776094957
How many subintervals have a interval of natural numbers?
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A range is specified by its largest and smallest elements, $x$ and $y$. If $x=n$, then there are $n$ choices for $y$, anything between $1$ and $n$ inclusive. If $x=(n-1)$, there are $n-1$ choices for $y$, and so on.
Continuing in this pattern, the number of ranges is $$n+(n-1)+(n-2)+\dots+2+1=n(n+1)/2.$$