tiny-tim said:Hi morbius27! Welcome to PF!
Try it for n = 4 first …
write out a list of all the closed intervals …
what do you get? when you count them, can you see a pattern?
The formula for the number of intervals contained in [1,n] is n(n+1)/2.
The formula can be proven using mathematical induction. First, show that the formula is true for n=1. Then, assume that the formula is true for some value of n. Finally, prove that the formula is also true for n+1. This will prove that the formula holds for all positive integers.
The number of intervals contained in [1,n] is significant in many areas of mathematics, such as combinatorics and calculus. It can also be used in real-world applications, such as calculating the number of possible outcomes in a game or the number of subintervals in a given range.
Yes, the formula can be extended to any interval [a,b] by simply replacing n with b-a in the formula. This will give the number of intervals contained in the range [a,b].
The formula assumes that the intervals are all of equal length, starting from 1 and ending at n. If the intervals are not of equal length or do not start from 1, then the formula may not be applicable. Additionally, the formula only applies to positive integers.