and we have to prove .
We proceed by induction:
.. so the base case holds.
suppose the statement holds for and let's prove it holds for :
(by induction hypotheses)
so the inductive step is done.
The succession [Xsubn] is defined as
Xsub0 = 0, Xsub(n+1) = Xsubn + n(n+1)
Show that Xsubn = (1/3)(n-1)n(n+1).
I am assuming I am not allowed to use what I am demonstring as part of the proof. No clue what do despite staring at it for 15 minutes.
But it is perfectly valid to show that what you are given is a solution to an equation by putting it into the equation.
For example, it would be very difficult to solve but it is very easy to show that x= 2 is a solutions: [tex]2^2- 6(2^2)- 3(2)- 6= 32- 24- 6- 2= 0.
Similarly, here, you are not asked to solve the recursion, just to show that satisfies it. And that is basically what Taluivren did.