Question 1092345
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<pre>
The key formula is 

{{{S[n]}}} = {{{1^3}}} + {{{2^3}}} + {{{3^3}}} +  . . . + {{{n^3}}} = {{{(n^2*(n+1)^2)/4}}}


For the proof see the lesson 
&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>
in this site.


Apply this formula for the sum 

{{{S[2n-1]}}} = {{{1^3}}} + {{{2^3}}} + {{{3^3}}} +  . . . + {{{(2n-1)^3}}}
 
= {{{((2n-1)^2*(2n)^2)/4}}} = {{{(2n-1)^2*n^2}}}.


Then calculate the sum over even natural numbers

{{{E[n]}}} = {{{2^3}}} + {{{4^3}}} + {{{6^3}}} +  . . . + {{{(2n-2)^3}}} = 

             {{{8*(1^3 + 2^3 + 3^3 + ellipsis + (n-1)^3)}}} = 

             {{{8*S[n-1]}}} = {{{8*((n-1)^2*n^2)/4}}} = {{{2*(n-1)^2*n^2}}}.


Now your sum is equal to  {{{S[2n-1]}}} - {{{E[n]}}} = {{{((2n-1)^2*n^2)}}} - {{{2*(n-1)^2*n^2}}} = {{{n^2*((2n-1)^2 - 2*(n-1)^2)}}} = {{{n^2*(4n^2 - 4n + 1 - 2n^2 + 4n -2)}}} = {{{n^2*(2n^2-1)}}} {{{2n^4 - n^2}}}.
</pre>

It is exactly what you want to prove. &nbsp;&nbsp;QED.



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To feel yourself freely with these/(with such) problems, you should be familiar with the method of Mathematical induction 
and with all associated standard problems around these theme.

There is a group of lessons in this site specially developed for this purpose.
These lessons are


&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 lessons are 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.




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