document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1022193>Question 1022193</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm having difficulty solving an exercise. <BR>\n" );
document.write( "The question is: <BR>\n" );
document.write( "Prove 3^n >=1+2*n. (proving by induction)<BR>\n" );
document.write( "Basis step is true        n=1 3^1>=1+2*1 <BR>\n" );
document.write( "I assume                  n=k is true  3^k>=1+2*k<BR>\n" );
document.write( "I now have to prove for   n=k+1 ;  3^k+1>=1+2(k+1)<BR>\n" );
document.write( "                                   3^k*3^1>=1+2*k+2<BR>\n" );
document.write( "This is where I get stuck.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #637978 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=637978\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assume induction hypothesis true for n = k, where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large k \ge 1\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 3^k \ge 1 + 2k\">\r<BR>\n" );
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document.write( "You want to prove that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 3^{k+1} \ge 1 + 2(k+1) = 3 + 2k\">\r<BR>\n" );
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document.write( "First, you can note that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 3^{k+1} = 3 \cdot 3^k\">, and this is greater than or equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 3(1+2k)\"> by induction hypothesis. So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 3^{k+1} \ge 3+6k\">\r<BR>\n" );
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document.write( "We also know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 3+6k \ge 3+2k\"> since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large k \ge 1\">.\r<BR>\n" );
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document.write( "Therefore\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 3^{k+1} \ge 3+6k \ge 3+2k\">\r<BR>\n" );
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document.write( "So the statement is true for n = k+1. </SPAN>\n" );
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