document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=527962>Question 527962</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove by induction that 2^n>=n^2. I was stocked at the induction step when 2^(k+1)>=2k^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #349102 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=349102\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The induction basically claims that if for some k > 1, if\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2^k \ge k^2\">, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2^{k+1} \ge (k+1)^2\">.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2^{k+1} = 2(2^k) \ge 2k^2\">, we can construct the inequality\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2^{k+1} \ge 2k^2 \ge (k+1)^2\">\r<BR>\n" );
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document.write( "The inequality <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2k^2 \ge (k+1)^2\"> is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2k^2 \ge k^2 + 2k + 1 \Rightarrow k^2 - 2k - 1 \ge 0\"> It factors to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (k-1)^2 - 2 \ge 0 \Rightarrow (k-1)^2 \ge 2\">, which is true for sufficiently large k (it doesn't hold for k = 0,1,2 but we can check these cases separately). </SPAN>\n" );
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