document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176890>Question 1176890</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Proof by induction. Imagine that we are going to prove by induction that:\r<BR>\n" );
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document.write( "(1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... + (1/sqrt(n)) >= sqrt(n), for all n E Z^+\r<BR>\n" );
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document.write( "Assume by the inductive step that:\r<BR>\n" );
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document.write( "(1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... + (1/sqrt(k)) >= sqrt(k), for some k E Z^+\r<BR>\n" );
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document.write( "Which of the following is a correct way of ending this proof?\r<BR>\n" );
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document.write( "a. (1/sqrt(1)) + (1/sqrt(2)) + ... + (1/sqrt(k)) >= sqrt(k) + (1/sqrt(k+1)) = sqrt(k+1) +1 >= sqrt(k+1)\r<BR>\n" );
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document.write( "b. (1/sqrt(1)) + (1/sqrt(2)) + ... + (1/sqrt(k)) >= sqrt(k+1) + (1/sqrt(k+1)) + (1/sqrt(k+1)) >= sqrt(k+1)\r<BR>\n" );
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document.write( "c. (1/sqrt(1)) + (1/sqrt(2)) + ... + (1/sqrt(k+1)) >= sqrt(k) + (1/sqrt(k+1)) = (sqrt(k)sqrt(k+1)+1)/sqrt(k+1)) >= (sqrt(k)sqrt(k)+1)/sqrt(k+1)) >= ((k+1)/sqrt(k+1)) = sqrt(k+1)\r<BR>\n" );
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document.write( "d. (1/sqrt(1)) + (1/sqrt(2)) + ... + (1/sqrt(k+1)) >= sqrt(k) + (1/sqrt(k)) = ((sqrt(k)sqrt(k+1))/sqrt(k)) >= ((k+1)/(sqrt(k+1))) = sqrt(k+1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #804905 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52780)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=804905\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "The proof (c) corresponds to the standard logic of the Mathematical induction method and corresponds \r<BR>\n" );
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document.write( "to the logic of the proof the necessary inequality.\r<BR>\n" );
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document.write( "==========\r<BR>\n" );
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document.write( "About the method of Mathematical induction, &nbsp;you may read from the lessons\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-geometric-progressions.lesson>Mathematical induction and geometric progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-for-sequences-other-than-arithmetic-or-geometric.lesson>Mathematical induction for sequences other than arithmetic or geometric</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Proving-inequalities-by-the-method-of-Mathematical-Induction.lesson>Proving inequalities by the method of Mathematical Induction</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/OVERVIEW-of-lessons-on-the-Method-of-Mathematical-induction.lesson>OVERVIEW of lessons on the Method of Mathematical induction</A>\r<BR>\n" );
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document.write( "Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lesson is the part of this online textbook under the topic <BR>\n" );
document.write( "<U>\"Method of Mathematical induction\"</U>.\r<BR>\n" );
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document.write( "Save the link to this textbook together with its description\r<BR>\n" );
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document.write( "Free of charge online textbook in ALGEBRA-II<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r<BR>\n" );
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document.write( "into your archive and use when it is needed.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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