perplexus.info :: Sequences : Verify and prove

Verify and prove (Posted on 2017-12-10)

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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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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