Quick question on induction problem

[tegra97]

I'm looking at an example in the book proving 1+3+5...+(2n-1)=n^2 by induction.

The example shows:
1+3+5+...(2n-1) + [(2n+1)-1] = (n^2)+[(2n+1)-1]
= (n^2) +2n+1
= (n+1)^2

My question is how did we get from [(2n+1)-1] to 2n+1???
Thanks

 

[stapel]

I would guess that there is a typo, since the (k+1)-th term should have followed the form of all the other terms:

. . . . .n = 1: 2(1) - 1 = 2 - 1 = 1
. . . . .n = 2: 2(2) - 1 = 4 - 1 = 3
. . . . .n = 3: 2(3) - 1 = 6 - 1 = 5
. . . . .n = 4: 2(4) - 1 = 8 - 1 = 7

So the k-th term would be:

. . . . .n = k: 2(k) - 1 = 2k - 1

...and the (k+1)-th term would be:

. . . . .n = k + 1: 2(k + 1) - 1 = 2k + 2 - 1 = 2k + 1

Then everything else should work correctly.

Eliz.

 

[Unco]

They meant to write:

I'm looking at an example in the book proving 1+3+5...+(2n-1)=n^2 by induction.

The example shows:
1+3+5+...(2n-1) + [2(n+1)-1] = (n^2)+[2(n+1)-1]
= (n^2) +2n+1
= (n+1)^2

My question is how did we get from [(2n+1)-1] to 2n+1???
Thanks