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Verify and prove (Posted on 2017-12-10) Difficulty: 4 of 5
Consider:
      (15 + 25)     +     (17 + 27)     =   2 *(1 + 2)4   
   (15 + 25 + 35)   +   (17 + 27 + 37)   =  2 *(1 + 2 + 3)4 
(15 + 25 + 35 + 45) + (17 + 27 + 37 + 47) = 2 *(1 + 2 + 3 + 4)4
       ...                  ...       and so on     ...

First, verify that both sides are equal for further increase in n,

then prove it.

No Solution Yet Submitted by Ady TZIDON    
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statements and references | Comment 3 of 5 |
Let s(k)=sum of k-powers of first n digits.

s(1)=n(n+1)/2=a.

s(5)=(4a^3-a^2)/3.

s(7)=(6a^4-4a^3+a^2)/3.

So s(5)+s(7)=2a^4 as desired.

Proof of each formula follows by induction.  You could use the calculus of differences to derive each, if you don't mind the tedium and you don't make a mistake, but it's easier and lots more fun to use the net.

Pascal got there early.

Bernoulli numbers play a part.

The end(?) is Faulhaber's Fabulous Formula.

  Posted by xdog on 2017-12-10 12:30:44
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