Question

Given the following relationship, prove (*)

Answer 1

Relationship:
\[(r+1)^2r^2(2r+1) = 10r^4 + 2r^2\]

To prove:
\[\sum_{r=1}^{n}r^4 = \frac{1}{30}n(n+1)(2n+1)(3n^2 + 3n - 1)\]

Answer 2

Thats a pretty weird relation, it only works for 0 and 1, but any other integers make it fail.

Answer 3

Are you sure thats right? one side is a 5th degree polynomial in r, and the other is a 4th degree.

Answer 4

Well, this is the question... @joemath314159

Answer 5

yeah that relationship seems a bit fishy

Answer 6

an easy way to prove that formula is by induction

Answer 7

it wouldn't use the relation, but that relation doesn't seem to work for many numbers..

Answer 8

\[r^2(r+1)^2(2r+1)=r^2(10r^2+2)\]\[r^2(2r^3 + 5r^2 +4r +1)=r^2(10r^2+2) \]\[r^2(2r^3 -5r^2 +4r -1) = 0\]

the only roots are r = 1, r= 0, r= 1/2

Answer 9

Looks like the question have limitation

Answer 10

if you want to prove the formula, use induction its nice :)

show it works for n = 0

then assume it works for n = k

now show that IF its true for n = k , it must be true for n=k+1

Answer 11

Ok... Thx