SOLUTION: Induction 1^3+3^3+5^3+...+(2n-1)^3 = 2n^4-n^2

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Question 1092345: Induction
1^3+3^3+5^3+...+(2n-1)^3 = 2n^4-n^2
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
The key formula is 

S%5Bn%5D = 1%5E3 + 2%5E3 + 3%5E3 +  . . . + n%5E3 = %28n%5E2%2A%28n%2B1%29%5E2%29%2F4


For the proof see the lesson 
    - Mathematical induction for sequences other than arithmetic or geometric
in this site.


Apply this formula for the sum 

S%5B2n-1%5D = 1%5E3 + 2%5E3 + 3%5E3 +  . . . + %282n-1%29%5E3
 
= %28%282n-1%29%5E2%2A%282n%29%5E2%29%2F4 = %282n-1%29%5E2%2An%5E2.


Then calculate the sum over even natural numbers

E%5Bn%5D = 2%5E3 + 4%5E3 + 6%5E3 +  . . . + %282n-2%29%5E3 = 

             8%2A%281%5E3+%2B+2%5E3+%2B+3%5E3+%2B+ellipsis+%2B+%28n-1%29%5E3%29 = 

             8%2AS%5Bn-1%5D = 8%2A%28%28n-1%29%5E2%2An%5E2%29%2F4 = 2%2A%28n-1%29%5E2%2An%5E2.


Now your sum is equal to  S%5B2n-1%5D - E%5Bn%5D = %28%282n-1%29%5E2%2An%5E2%29 - 2%2A%28n-1%29%5E2%2An%5E2 = n%5E2%2A%28%282n-1%29%5E2+-+2%2A%28n-1%29%5E2%29 = n%5E2%2A%284n%5E2+-+4n+%2B+1+-+2n%5E2+%2B+4n+-2%29 = n%5E2%2A%282n%5E2-1%29 2n%5E4+-+n%5E2.

It is exactly what you want to prove.   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

    - 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
    - OVERVIEW of lessons on the Method of Mathematical induction

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Method of Mathematical induction".


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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