document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=438225>Question 438225</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use the principal mathematical induction to show that the statement is true for all natural numbers n. \r<BR>\n" );
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document.write( "9+18+27+...+9n= 9n(n+1)/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #303122 by </B><A HREF=/tutors/aboutme.mpl?userid=Gogonati><B>Gogonati(855)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Gogonati><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=Gogonati><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=303122\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To prove a statement by mathematical induction we use three steps:\r<BR>\n" );
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document.write( "Step 1. We prove that the statement 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=9=9%2A1%281%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"9=9%2A1%281%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=18%2F2 ALIGN=MIDDLE  ALT=\"9=18%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=9 ALIGN=MIDDLE  ALT=\"9=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 2. Assume that the statement is true for n=k.\r<BR>\n" );
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document.write( " 9+18+27+...+9k = 9k(k+1)/2.\r<BR>\n" );
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document.write( "Step 3.We prove that the statement is true for n=k+1.\r<BR>\n" );
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document.write( " 9+18+27+...+9k+9(k+1) = 9(k+1)(k+2)/2.\r<BR>\n" );
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document.write( "Substitute the assuming result in step 2 to the step 3 we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9k%28k%2B1%29%2F2%2B9%28k%2B1%29=9k%28k%2B1%292%2B18%28k%2B1%29%2F2=%28k%2B1%29%289k%2B18%29%2F2=9%28k%2B1%29%28k%2B2%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, thus \r<BR>\n" );
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document.write( "the statement is true. </SPAN>\n" );
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