document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=226228>Question 226228</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Trying to help my friend with his algebra, and I'm stumped as to what they're going for on this one:<BR>\n" );
document.write( "Explain why:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29-sqrt%28n%2B1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29-sqrt%28n%2B1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is true for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3E=1 ALIGN=MIDDLE  ALT=\"n%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > two different ways.<BR>\n" );
document.write( "a) Provide two numerical values.<BR>\n" );
document.write( "b) Show that is it true in general.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #168510 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=168510\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The equation should be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Make sure that you have the correct equation.\r<BR>\n" );
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document.write( "a) Let's show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is true for the values <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=2 ALIGN=MIDDLE  ALT=\"n=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's show the equation is true for n=1:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%281%29%2Bsqrt%281-1%29%29%28sqrt%281%29-sqrt%281-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%281%29%2Bsqrt%281-1%29%29%28sqrt%281%29-sqrt%281-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%281%29%2Bsqrt%280%29%29%28sqrt%281%29-sqrt%280%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%281%29%2Bsqrt%280%29%29%28sqrt%281%29-sqrt%280%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B0%29%281-0%29=1 ALIGN=MIDDLE  ALT=\"%281%2B0%29%281-0%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of 1 to get 1. Take the square root of 0 to get 0.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%281%29=1 ALIGN=MIDDLE  ALT=\"%281%29%281%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=1 ALIGN=MIDDLE  ALT=\"1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is true when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's show the equation is true for n=2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2Bsqrt%282-1%29%29%28sqrt%282%29-sqrt%282-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%282%29%2Bsqrt%282-1%29%29%28sqrt%282%29-sqrt%282-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2Bsqrt%281%29%29%28sqrt%282%29-sqrt%281%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%282%29%2Bsqrt%281%29%29%28sqrt%282%29-sqrt%281%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2B1%29%28sqrt%282%29-1%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%282%29%2B1%29%28sqrt%282%29-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of 1 to get 1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2Asqrt%282%29-sqrt%282%29%2Bsqrt%282%29%2B1%28-1%29=1 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2Asqrt%282%29-sqrt%282%29%2Bsqrt%282%29%2B1%28-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-sqrt%282%29%2Bsqrt%282%29-1=1 ALIGN=MIDDLE  ALT=\"2-sqrt%282%29%2Bsqrt%282%29-1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=1 ALIGN=MIDDLE  ALT=\"1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29-sqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is true when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=2 ALIGN=MIDDLE  ALT=\"n=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Now let's show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29%2Bsqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29%2Bsqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is true for any value of 'n' such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3E=1 ALIGN=MIDDLE  ALT=\"n%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29%2Bsqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29%2Bsqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28n%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sqrt%28n-1%29 ALIGN=MIDDLE  ALT=\"y=sqrt%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2By%29%28x-y%29=1 ALIGN=MIDDLE  ALT=\"%28x%2By%29%28x-y%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28n%29 ALIGN=MIDDLE  ALT=\"sqrt%28n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with 'x' and replace each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28n-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with 'y'\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-xy%2Bxy-y%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2-xy%2Bxy-y%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-y%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2-y%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%29%5E2-%28sqrt%28n-1%29%29%5E2=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%29%5E2-%28sqrt%28n-1%29%29%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28n%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sqrt%28n-1%29 ALIGN=MIDDLE  ALT=\"y=sqrt%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-%28n-1%29=1 ALIGN=MIDDLE  ALT=\"n-%28n-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square each square root to eliminate them (note: because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3E=1 ALIGN=MIDDLE  ALT=\"n%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-1%3E=0 ALIGN=MIDDLE  ALT=\"n-1%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that each radicand is positive. So we can avoid absolute values here).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-n%2B1=1 ALIGN=MIDDLE  ALT=\"n-n%2B1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=1 ALIGN=MIDDLE  ALT=\"1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So this shows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29%2Bsqrt%28n-1%29%29=1 ALIGN=MIDDLE  ALT=\"%28sqrt%28n%29%2Bsqrt%28n-1%29%29%28sqrt%28n%29%2Bsqrt%28n-1%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is true for any value of 'n' such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3E=1 ALIGN=MIDDLE  ALT=\"n%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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