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 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 No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 Why it works Comment 5 of 5 |

Start with RHS = 2(n/2 (n + 1))^4 = 1/8 (n^2 + n)^4

Then LHS, considering each bracket separately:

Sum (1 to n) n^5 = 1/6 (n^2 + n)^3 - 1/12 (n^2 + n)^2

Sum (1 to n) n^7 = 1/8 (n^2 + n)^4 - 1/6 (n^2 + n)^3 + 1/12 (n^2 + n)^2

So the n^5 term exactly cancels out of the n^7 term, leaving 1/8 (n^2 + n)^4, as was to be shown. Very neat.

 Posted by broll on 2017-12-10 21:33:38

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