SOLUTION: 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)^

Question 1080517: Using mathematical induction, show its true for all natural #'s n
+ =
So far I got
Proof n=1
=
1 = 3/3
1 = 1
so its true for n=1
next assume n = k, so
+ =
then we must prove n = k+1
+ + =
we do some replacing with our n = k

+ =
we have our left hand side that we must match to our right hand side, but this is where my math get's messy when I go to simplify,
+ =
+ =
now I'm left with a +4k^2 and I'm unsure what method to use here, or maybe I did the first steps wrong? If I factor the k(2k-1)(2k+1) I get a 4k^3-k and I thought maybe multiply by (3/3) to get everything as denominator of 3, but it doesn't really help me understand what I should do next for certain. Any help would be greatly appreciated
Answer by ikleyn(52778) (Show Source): You can put this solution on YOUR website!
.
I start reproducing your post; then make some correction of your writing and then complete the proof.

Using mathematical induction, show its true for all natural #'s n 
+  = 

So far I got 
Proof n=1
 = 
1 = 3/3
1 = 1
so its true for n=1

next assume n = k, so 

+  = 

then we must prove n = k+1

+  +  =         <<<---Corrected

we do some replacing with our n = k 
 
  +  =                        <<<---Corrected

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

  +  =                        <<<---Corrected

  +  =                            <<<---Corrected

Below I continue the proof.

Now let us transform the left side step by step:

 +  = 

  +  = 

 = 

 = .   (*)

Now notice that  = ,  and therefore you can continue the chain of equalities (*) in this way

=  = ,

and this is EXACTLY our RIGHT side.

The proof is completed.

-----------
There is a free of charge textbook in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

There is a chapter in this textbook Method of Mathematical induction.

There are the lessons in this chapter
    - 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

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

Happy learning !!

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