document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=170325>Question 170325</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use mathematical induction to prove:  4^n+1 + 5^2n-1 is divisible by 21 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #125771 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=125771\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: I'm assuming that the assumption holds 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;\" >\r<BR>\n" );
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document.write( "Step 1) \r<BR>\n" );
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document.write( "Prove for n=1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28n%2B1%29+%2B+5%5E%282n-1%29=4%5E%281%2B1%29+%2B+5%5E%282%281%29-1%29=4%5E%282%29+%2B+5%5E%281%29=16%2B5=21 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > which is divisible by 21\r<BR>\n" );
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document.write( "Step 2) \r<BR>\n" );
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document.write( "Assume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28k%2B1%29+%2B+5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28k%2B1%29+%2B+5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by 21 (ie assume kth term is divisible by 21)\r<BR>\n" );
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document.write( "Step 3) \r<BR>\n" );
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document.write( "Prove true for k+1 term\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28k%2B1%29+%2B+5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28k%2B1%29+%2B+5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the assumed portion\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28k%2B1%2B1%29+%2B+5%5E%282%28k%2B1%29-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28k%2B1%2B1%29+%2B+5%5E%282%28k%2B1%29-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in k+1 for every k\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28k%2B1%2B1%29+%2B+5%5E%282k%2B2-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28k%2B1%2B1%29+%2B+5%5E%282k%2B2-1%29\"   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=4%2A4%5E%28k%2B1%29+%2B+5%5E2%2A5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%2A4%5E%28k%2B1%29+%2B+5%5E2%2A5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the exponent\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A4%5E%28k%2B1%29+%2B+25%2A5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%2A4%5E%28k%2B1%29+%2B+25%2A5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 5 to get 25\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A4%5E%28k%2B1%29+%2B+4%2A5%5E%282k-1%29%2B21%2A5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%2A4%5E%28k%2B1%29+%2B+4%2A5%5E%282k-1%29%2B21%2A5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up 25 to get 4+21\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%284%5E%28k%2B1%29+%2B+5%5E%282k-1%29%29%2B21%2A5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%284%5E%28k%2B1%29+%2B+5%5E%282k-1%29%29%2B21%2A5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF 4\r<BR>\n" );
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document.write( "Since we're assuming that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28k%2B1%29+%2B+5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28k%2B1%29+%2B+5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by 21, this means that  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28k%2B1%29+%2B+5%5E%282k-1%29=21m ALIGN=MIDDLE  ALT=\"4%5E%28k%2B1%29+%2B+5%5E%282k-1%29=21m\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some integer \"m\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2821m%29%2B21%2A5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%2821m%29%2B21%2A5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28k%2B1%29+%2B+5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28k%2B1%29+%2B+5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with 21m\r<BR>\n" );
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document.write( "Now let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"n=5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is an integer)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2821m%29%2B21n ALIGN=MIDDLE  ALT=\"4%2821m%29%2B21n\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E%282k-1%29 ALIGN=MIDDLE  ALT=\"5%5E%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with \"n\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%284m%2Bn%29 ALIGN=MIDDLE  ALT=\"21%284m%2Bn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF 21\r<BR>\n" );
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document.write( "Now let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=j=4m%2Bn ALIGN=MIDDLE  ALT=\"j=4m%2Bn\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the expression becomes\r<BR>\n" );
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document.write( "Since 21 is a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21j ALIGN=MIDDLE  ALT=\"21j\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this shows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28k%2B1%2B1%29+%2B+5%5E%282%28k%2B1%29-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28k%2B1%2B1%29+%2B+5%5E%282%28k%2B1%29-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by 21. \r<BR>\n" );
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document.write( "So this proves that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28n%2B1%29+%2B+5%5E%282n-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28n%2B1%29+%2B+5%5E%282n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by 21 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;\" >\r<BR>\n" );
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