document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=914968>Question 914968</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help.<BR>\n" );
document.write( "(sqrt(n)+sqrt(n-1))(sqrt(n)-sqrt(n-1))=1 for n>=1<BR>\n" );
document.write( "a.provide two numerical examples illustrating its validity.<BR>\n" );
document.write( "b.show the statement is true in general.<BR>\n" );
document.write( "c.what is the difference between using numerical values to show that something is true and showing in general that something is true? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #555366 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=555366\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. n = 1 --> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (1+0)(1-0) = 1\"><BR>\n" );
document.write( "n = 10 --> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (\sqrt{10} + 3)(\sqrt{10} - 3) = 1\"> (you can easily check this)\r<BR>\n" );
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document.write( "b. Difference of squares or FOIL:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (\sqrt{n} + \sqrt{n-1})(\sqrt{n} - \sqrt{n-1}) = (\sqrt{n})^2 - (\sqrt{n-1})^2 = n - (n-1) = 1\">\r<BR>\n" );
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document.write( "c. Showing the statement holds in general is a much stronger proof - checking a few cases usually does not constitute a valid proof. In fact, checking a few cases is widely considered invalid unless you can show that you checked all possible cases, or all possible cases reduce to ones you've already checked.\r<BR>\n" );
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document.write( "For example, to show that n^6 leaves a remainder of 0 or 1 when divided by 7, you only need to check n = 0,1,2,...,6 (do you see why?). However this is true by Fermat's little theorem. </SPAN>\n" );
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