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

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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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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 ;)
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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