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Tuesday, September 8, 2015

Prove that: n+1/(n+(n+2))=n, for all nN.

Prove that: n+1n+n+2=n, for all nN.

My solution:

Step 1:

First, note that the expression inside the floor function on the left can be rewritten such that we have:

n+1n+n+2

=n+1n+n+2nn+2nn+2

=n+nn+2n(n+2)

=nnn+22

=n+n+2n2

=n+n+22

Step 2:

Next, note that for all nN, 4n>2 is always true. Algebraically manipulating it such that we get:

n+4n+4>2+n+4

(n+2)2>n+2

n+2>n+2

Therefore we have:

n+n+2>n+n+2, which is

2n+2>n+n+2

Step 3:

Note that we can set:

n+n<n+n+2<2n+2

which is

2n<n+n+2<2(n+1)

n<n+n+22<n+1

Thus, we can conclude at this juncture that n+n+22=n+1n+n+2=n and we're hence done.

1 comment:

  1. i dont Think that you can draw that conclusion simple because sqrt(x) is not necessarily an integer. för example if x € (2.3 , 3.3) then the integer part of x could either 2 or 3.
    Sébastien

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