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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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Solution Classic induction | Comment 4 of 5 |
Show the pattern works for n=1
1^5+7^5=2=2*(1)^4

Assume it works for n-1: 
(1^5+2^5+...+(n-1)^5)+(1^7+2^7+...+(n-1)^7)=2*(1+2+...+(n-1))^4
Show it works for n:
(1^5+2^5+...+(n)^5)+(1^7+2^7+...+(n)^7)=2*(1+2+...+(n))^4
The difference being

n^7+n^5 = 2[(1+2+...+(n-1))^4 - (1+2+...+(n))^4]
=2[[n(n+1)/2]^4 - [[(n-1)n/2]^4]
=n^4((n+1)^4-(n-1)^4)/8
=n^4(8n^3+8n)/8
=n^7+n^5

  Posted by Jer on 2017-12-10 19:38:26
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