Question 1129620
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<U>The base of induction</U>


<pre>
    At n= 1  n^3 + 2n = 1^3 + 2*1 = 3  is divisible by 3.

    Thus the base of induction is valid.
</pre>


<U>The induction step</U>


<pre>
    Let assume that  P(n) = n^3 + 2n is divisible by 3,



    Then  P(n+1) = (n+1)^3 + 2*(n+1) = n^3 + 3n^2 + 3n + 1 + 2n + 2 = 

                 = (re-group) = (n^3 + 2n) + (3n^2 + 3n + 3) = P(n) + 3*(n^2 + n +1).



    So, P(n+1) is the sum of P(N) and the other addend, which is multiple of 3.


    Thus, if P(n) is divisible by n, then P(n+1) is divisible by 3, too.


    The inductive step is proven to be valid.
</pre>


Hence, according to the Mathematical induction principle, the statement is true for all positive integer n.


The proof is completed.


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On the method of Mathematical induction see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-geometric-progressions.lesson>Mathematical induction and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A 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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A 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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A 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>


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


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



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.