document.write( "Question 1176890: Proof by induction. Imagine that we are going to prove by induction that:\r
\n" ); document.write( "\n" ); document.write( "(1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... + (1/sqrt(n)) >= sqrt(n), for all n E Z^+\r
\n" ); document.write( "\n" ); document.write( "Assume by the inductive step that:\r
\n" ); document.write( "\n" ); document.write( "(1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... + (1/sqrt(k)) >= sqrt(k), for some k E Z^+\r
\n" ); document.write( "\n" ); document.write( "Which of the following is a correct way of ending this proof?\r
\n" ); document.write( "\n" ); 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
\n" ); document.write( "\n" ); 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
\n" ); document.write( "\n" ); 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
\n" ); document.write( "\n" ); 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)
\n" ); document.write( "

Algebra.Com's Answer #804905 by ikleyn(52778)\"\" \"About 
You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The proof (c) corresponds to the standard logic of the Mathematical induction method and corresponds \r
\n" ); document.write( "\n" ); document.write( "to the logic of the proof the necessary inequality.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "==========\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "About the method of Mathematical induction,  you may read from the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction for sequences other than arithmetic or geometric\r
\n" ); document.write( "\n" ); document.write( "    - Proving inequalities by the method of Mathematical Induction\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on the Method of Mathematical induction\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this online textbook under the topic
\n" ); document.write( "\"Method of Mathematical induction\".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "
\n" );