document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=923732>Question 923732</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For which positive integers n is 11n + 17 &#8804; 2n? Prove the conjecture you made using<BR>\n" );
document.write( "mathematical induction. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #560309 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=560309\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> No positive integer n satisfies 11n + 17 <= 2n (this is equivalent to n <= -17/9).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Proof: n = 1 --> 11*1 + 17 <= 2*1, which is not true. Suppose that some k >= 1 does not satisfy inequality, which occurs iff 11k + 17 > 2k. We wish to show that k+1 also does not satisfy the inequality, i.e. we wish to show\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 11(k+1) + 17 > 2(k+1)\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 11k + 17 + 11 > 2k + 2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "However by hypothesis, 11k+17 > 2k, so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large 11k + 17 + 11 > 2k + 11 > 2k + 2\">, which is a true statement, so k+1 also does not satisfy 11n + 17 <= 2n. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );