Question 957377
Consider {{{ ( sqrt(n) + sqrt(n-1)) ( sqrt(n) - sqrt(n-1)) }}} =1 for {{{n >or= 1}}} 

-Provide two numerical examples illustrating validity.
<pre>
Let n = 1

{{{sqrt(1)sqrt(1)-sqrt(1)sqrt(1-1)+sqrt(1-1)sqrt(1)-sqrt(1-1)sqrt(1-1)}}}{{{""=""}}}{{{1}}}

{{{(1)(1)-(1)sqrt(0)-sqrt(0)sqrt(1)+sqrt(0)sqrt(0)}}}{{{""=""}}}{{{1}}}

{{{1-(1)(0)+(0)(1)-(0)(0)}}}{{{""=""}}}{{{1}}}
 
{{{1-0+0-0}}}{{{""=""}}}{{{1}}}

{{{1}}}{{{""=""}}}{{{1}}}

----------------------

Let n = 2

{{{sqrt(2)sqrt(2)-sqrt(2)sqrt(2-1)+sqrt(2-1)sqrt(2)-sqrt(2-1)sqrt(2-1)}}}{{{""=""}}}{{{1}}}

{{{2-sqrt(2)sqrt(1)-sqrt(1)sqrt(2)+sqrt(1)sqrt(1)}}}{{{""=""}}}{{{1}}}

{{{2-sqrt(2)(1)+(1)sqrt(2)-(1)(1)}}}{{{""=""}}}{{{1}}}
 
{{{2-sqrt(2)+sqrt(2)-1}}}{{{""=""}}}{{{1}}}

{{{2-cross(sqrt(2))+cross(sqrt(2))-1}}}{{{""=""}}}{{{1}}}

{{{2-1}}}{{{""=""}}}{{{1}}}

{{{1}}}{{{""=""}}}{{{1}}}

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</pre>
-Show the statement is true in general. 
<pre>
{{{ ( sqrt(n) + sqrt(n-1)) ( sqrt(n) - sqrt(n-1)) }}}{{{""=""}}}{{{1}}}

Use FOIL:

{{{sqrt(n)sqrt(n)-sqrt(n)sqrt(n-1)-sqrt(n-1)sqrt(n)-sqrt(n-1)sqrt(n-1)}}}{{{""=""}}}{{{1}}}

{{{sqrt(n)sqrt(n)-cross(sqrt(n)sqrt(n-1))+cross(sqrt(n-1)sqrt(n))-sqrt(n-1)sqrt(n-1)}}}{{{""=""}}}{{{1}}}

{{{sqrt(n)sqrt(n)-sqrt(n-1)sqrt(n-1)}}}{{{""=""}}}{{{1}}}

{{{n-(n-1)}}}{{{""=""}}}{{{1}}}

{{{n-n+1}}}{{{""=""}}}{{{1}}}

{{{1}}}{{{""=""}}}{{{1}}}

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</pre>
-What is the difference between using numerical values to show that 
something is true and showing in general that something is true?
<pre>
-When we use numerical values to show that something is true, we show it
ONLY for those particular numerical values and no others.

-What we show in general that something is true, we show it for all the
values stated, in this case for {{{n >= 1}}} 

--------------------
Edwin</pre>