Proving a formula for the number of intervals contained in [1,n]

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Homework Help Overview

The discussion revolves around proving a formula for the number of closed intervals with integer endpoints contained within the interval [1,n], where n is a natural number. The original poster attempts to understand how to formally prove the claim regarding the sum of intervals.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants suggest starting with specific cases, such as n = 4, to identify patterns in the intervals. There is mention of using induction as a potential method for proof.

Discussion Status

The discussion is ongoing, with some participants providing guidance on exploring specific cases to uncover patterns. There is an indication that the original poster found success in identifying a pattern after following suggestions, but no consensus or final proof has been reached.

Contextual Notes

Participants are encouraged to explore the problem through examples and patterns, indicating a focus on understanding rather than providing direct solutions. The nature of the problem suggests a need for careful consideration of definitions and assumptions regarding intervals.

morbius27
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Homework Statement


Prove a simple formula for the number of closed intervals with integer endpoints contained in the interval [1,n] (including one point intervals), where n is a natural number.


The Attempt at a Solution


I know that the formula ends up being the sum of i from i=1 to i=n (this makes sense just from inspection) but I have no idea how to actually go about proving the claim.
 
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Welcome to PF!

Hi morbius27! Welcome to PF! :wink:

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? :smile:​
 
if you know the solution but want to prove it, use induction
 


tiny-tim said:
Hi morbius27! Welcome to PF! :wink:

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? :smile:​

Thanks Tiny Tim! I got the solution after doing what you said and got the pattern; it worked beautifully after that ;)
 

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