SOLUTION: (1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... + (1/sqrt(n)) >= sqrt(n), for all n E Z^+ Assume by the inductive step that: (1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... +

Question 1176891: (1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... + (1/sqrt(n)) >= sqrt(n), for all n E Z^+

Assume by the inductive step that:

(1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... + (1/sqrt(k)) >= sqrt(k), for some k E Z^+

Which of the following is a correct way of ending this proof?

a. (1/sqrt(1)) + (1/sqrt(2)) + ... + (1/sqrt(k)) >= sqrt(k) + (1/sqrt(k+1)) = sqrt(k+1) +1 >= sqrt(k+1)

b. (1/sqrt(1)) + (1/sqrt(2)) + ... + (1/sqrt(k)) >= sqrt(k+1) + (1/sqrt(k+1)) + (1/sqrt(k+1)) >= sqrt(k+1)

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)

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)
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.

The proof (c) corresponds to the standard logic of the Mathematical induction method and corresponds
to the logic of the proof the associated inequality.

==========

About the method of Mathematical induction, you may read from the lessons
    - Mathematical induction and arithmetic progressions
    - Mathematical induction and geometric progressions
    - Mathematical induction for sequences other than arithmetic or geometric
    - Proving inequalities by the method of Mathematical Induction
    - OVERVIEW of lessons on the Method of Mathematical induction

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Method of Mathematical induction".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

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