Question 1066050
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The sum in the left side is equal to the sum of terms {{{(i*(i^2-1))}}} from i = 1 to i = N, 

which is the same as the sum of terms {{{i^3 - i}}} from i = 1 to i = N,


which in turn is the difference  A - B, where


A = sum {{{i^3}}} from i= 1 to i=N,     (1)

and

B = sum {{{i}}} from i= 1 to i=N.       (2)


Regarding the sum (1) see the lesson 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://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.


Regarding the sum (2) see the lesson 

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

in this site.
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Also, 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>".