# N equals N plus one

Theorem: n=n+1

Proof:

(n+1)^2 = n^2 + 2*n + 1

Bring 2n+1 to the left:

(n+1)^2 - (2n+1) = n^2

Substract n(2n+1) from both sides and factoring, we have:

(n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)

Adding 1/4(2n+1)^2 to both sides yields:

(n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2

This may be written:

[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2

Taking the square roots of both sides:

(n+1) - 1/2(2n+1) = n - 1/2(2n+1)

Add 1/2(2n+1) to both sides:

n+1 = n

Proof:

(n+1)^2 = n^2 + 2*n + 1

Bring 2n+1 to the left:

(n+1)^2 - (2n+1) = n^2

Substract n(2n+1) from both sides and factoring, we have:

(n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)

Adding 1/4(2n+1)^2 to both sides yields:

(n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2

This may be written:

[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2

Taking the square roots of both sides:

(n+1) - 1/2(2n+1) = n - 1/2(2n+1)

Add 1/2(2n+1) to both sides:

n+1 = n

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