Hello,

What we know (from the induction hypothsesis):

1^4 + 2^4 + 3^4 + .... + (n-1)^4 = (1/30)(6(n-1)^5 + 15(n-1)^4 + 10(n-1)^3 - (n-1)).

What we want to prove:

1^4 + 2^4 + 3^4 + .... + n^4 = (1/30)(6n^5 + 15n^4 + 10n^3 - n)

So, all we have to prove is:

-n^4=(1/30)(6(n-1)^5 + 15(n-1)^4 + 10(n-1)^3 - (n-1))-(1/30)(6n^5 + 15n^4 + 10n^3 - n).

Bye