You are not logged in.
#1 2005-10-23 07:52:52
- jd
- Member
Offline
Prove that 2n natural numbers is n(2n+1)
Could someone help I'm not sure how to do this.
"Show that the sum of the first 2n natural numbers is n(2n+1)"
It's a question from the heinemann c1 book.
I would ask the teacher again but it's half term and I don't know his phone number.
good day
Last edited by jd (2005-10-23 07:55:31)
#2 2005-10-23 10:00:55
- kylekatarn
- Power Member
Offline
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=◊(n-1)+n
◊n=◊(n-2)+(n-1)+n
◊n=◊(n-3)+(n-2)+(n-1)+n
◊n=◊(n-4)+(n-3)+(n-2)+(n-1)+n
.
.
.
◊n=(n-n+0)+(n-n+1)+...+(n-3)+(n-2)+(n-1)+n
◊n=0+1+2+...+n-3+n-2+n-1+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+0-n+1-n+2+...-3-2-1
◊n=[n²+n]-(n-0+n-1+n-2+...+3+2+1)
◊n=[n²+n]-(1+2+3+...+(n-2)+(n-1)+(n-0))
◊n=[n²+n]-(1+2+3+...+(n-2)+(n-1)+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 2005-10-24 02:49:47
- kylekatarn
- Power Member
Offline
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:
◊n=1+2+3+...+n
I just reduced your problem to a simpler version. Instead of trying to build an expression for the sum of first 2n naturals, I got the expression for the sum of first N naturals, ◊n. After that you just plug 2n: ◊(2n) and get the expression.
Think '◊n' as ◊(n) or f(n). There is nothing special about ◊.. It's just a symbol I usually associate with the 'function' that sums the 1st N naturals.
Now read my post 2 or 3 more times and try to catch up with a sheet of paper, line by line.
#5 2005-10-24 14:09:46
- ganesh
- Moderator
Offline
Re: Prove that 2n natural numbers is n(2n+1)
jd wrote:
I only started a level is there a easier way? I think I saw "S=n/2[2a + (n-1)d]" when the teacher did it.
You can think of a sum of n natural numbers as an Arithmetic Progression and use this formula.
S = n/2[2a + (n-1)d]
Here, n = n, a = 1, d = 1
Therefore, S = n/2[2 + (n-1)1]
S = n(n+1)/2
Character is who you are when no one is looking.
#6 2005-10-25 05:23:30
- mathsyperson
- Moderator
Offline
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.
ganesh: kylekatarn "would not accept that as a full demonstration".
Why did the vector cross the road?
It wanted to be normal.
#7 2005-10-25 05:30:19
- kylekatarn
- Power Member
Offline
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.
Again, it was NOT my intention to report that post.
#8 2005-10-25 05:48:18
- mathsyperson
- Moderator
Offline
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.