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You are not logged in. #1 20051023 07:52:52
Prove that 2n natural numbers is n(2n+1)Could someone help I'm not sure how to do this. Last edited by jd (20051023 07:55:31) #2 20051023 10:00:55
Re: Prove that 2n natural numbers is n(2n+1)Code:n let ◊n = 1+2+3+...+n = ∑ k k=1 ◊0=0 ◊1=1 //finding a formula is easy ◊n=◊(n1)+n ◊n=◊(n2)+(n1)+n ◊n=◊(n3)+(n2)+(n1)+n ◊n=◊(n4)+(n3)+(n2)+(n1)+n . . . ◊n=(nn+0)+(nn+1)+...+(n3)+(n2)+(n1)+n ◊n=0+1+2+...+n3+n2+n1+n ` n+1 terms ´ ◊n=[n+n+n+....+n+n]+(n+0)+(n+1)+(n+2)+...+(3)+(2)+(1) ◊n=[n²+n]+(n+0)+(n+1)+...+(3)+(2)+(1) ◊n=[n²+n]n+0n+1n+2+...321 ◊n=[n²+n](n0+n1+n2+...+3+2+1) ◊n=[n²+n](1+2+3+...+(n2)+(n1)+(n0)) ◊n=[n²+n](1+2+3+...+(n2)+(n1)+n) ◊n=[n²+n]◊n 2◊n=n²+n n²+n ◊n =  2 //great! now we have a formula for ◊n. //but the sum of the first 2n natural numbers is 2.n ∑ k k=1 //wait..don't we have a formula for that? Yes! λ ∑ k = ◊(λ) k=1 2.n (2n)²+2n (2n)(2n+1) n(2n+1) ∑ k = ◊(2n) =  =  =  >>>>> Done! k=1 2 2 2 #4 20051024 02:49:47
Re: Prove that 2n natural numbers is n(2n+1)If you read the demonstration carefully, you will see right at the beginning the ◊ 'operator' as I define it: #5 20051024 14:09:46
Re: Prove that 2n natural numbers is n(2n+1)
You can think of a sum of n natural numbers as an Arithmetic Progression and use this formula. Character is who you are when no one is looking. #6 20051025 05:23:30
Re: Prove that 2n natural numbers is n(2n+1)kylekatarn: You should only report abusive or spammy posts, things you think should be deleted because they are inappropriate. If you think ganesh's explanation isn't full enough, just say so within the topic. Why did the vector cross the road? It wanted to be normal. #7 20051025 05:30:19
Re: Prove that 2n natural numbers is n(2n+1)Sorry the whole mess but THAT was my REPLY! I clicked REPORT instead of QUOTE. #8 20051025 05:48:18
Re: Prove that 2n natural numbers is n(2n+1)Well, fair enough, then. Don't worry, we all make mistkaes. Why did the vector cross the road? It wanted to be normal. 