Question 1080517
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<U>I start reproducing your post; then make some correction of your writing and then complete the proof.</U>


<pre>
Using mathematical induction, show its true for all natural #'s n 
{{{ 1^2+3^2+5^2+ ...}}}+ {{{ (2n-1)^2 }}} = {{{ (n(2n-1)(2n+1))/3 }}}

So far I got 
Proof n=1
{{{ (2(1)-1)^2 }}} = {{{ (1(2(1)-1)(2(1)+1))/3 }}}
1 = 3/3
1 = 1
so its true for n=1

next assume n = k, so 

{{{ 1^2+3^2+5^2+ ...}}}+ {{{ (2k-1)^2 }}} = {{{ (k(2k-1)(2k+1))/3 }}}

then we must prove n = k+1

{{{ 1^2+3^2+5^2+ ...}}}+ {{{ (2k-1)^2 }}} + {{{ (2(k+1)-1)^2 }}} = {{{ ((k+1)*(2(k+1)-1)(2(k+1)+1))/3 }}}        <<<---Corrected

we do some replacing with our n = k 
 
 {{{ (k(2k-1)(2k+1))/3 }}} + {{{ (2(k+1)-1)^2 }}} = {{{ ((k+1)*(2k+1)(2k+3))/3 }}}                       <<<---Corrected

we have our left hand side that we must match to our right hand side             

 {{{ (k(2k-1)(2k+1))/3 }}} + {{{ (2(k+1)-1)^2 }}} = {{{ ((k+1)*(2k+1)(2k+3))/3 }}}                       <<<---Corrected

 {{{ (k(2k-1)(2k+1))/3 }}} + {{{ (2k+1)^2 }}} = {{{ ((k+1)*(2k+1)(2k+3))/3 }}}                           <<<---Corrected
</pre>

<U>Below I continue the proof</U>.


<pre>
Now let us transform the left side step by step:

{{{ (k(2k-1)(2k+1))/3 }}} + {{{ (2k+1)^2 }}} = 

{{{ (k(2k-1)(2k+1))/3}}}  + {{{(3*(2k+1)^2)/3 }}} = 

{{{ ((2k+1)*(k*(2k-1) + 3*(2k+1)))/3}}} = 

{{{((2k+1)*(2k^2-k + 6k + 3))/3}}} = {{{((2k+1)*(2k^2 + 5k + 3))/3}}}.   (*)

Now notice that {{{2k^2 + 5k + 3}}} = {{{(k+1)*(2k+3)}}},  and therefore you can continue the chain of equalities (*) in this way

= {{{((2k+1)*(k+1)*(2k+3))/3}}} = {{{((k+1)*(2k+1)*(2k+3))/3}}},

and this is <U>EXACTLY</U> our <U>RIGHT</U> side.
</pre> 

The proof is completed.



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There is a free of charge textbook 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>.


There is a chapter in this textbook <U>Method of Mathematical induction</U>.


There are the lessons in this chapter 

&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>


that might be interested to you,  especially the lesson marked (**).



Happy learning !!