Again, it was NOT my intention to report that post.]]>

ganesh: kylekatarn "would not accept that as a full demonstration".

]]>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

◊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.

]]>◊0=0

◊1=1

mean?

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.

thanks, JD

```
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
```

]]>"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