Differentiating a term n+1 times

[aaaa202]

Differentiating the term:

(x²-1)u' n+1 times gives (according to my book):

(x²-1)u⁽ⁿ⁺²⁾ + 2x(n+1)u⁽ⁿ⁺¹⁾ + n(n+1)u⁽ⁿ⁾

Now I see how the first two parts arise. However, I don't really understand the last one - or more specifically I don't understand how you find that uⁿ appears exactly n(n+1) times. It is pretty clear that every uⁿ comes from when you have 2xuⁿ and then invoke the product rule, but how do you know that the sum of all these contributions (2 + 4 + 6 + 8 + 10).. sum up to n(n+1)?

 

[micromass]

You could prove it by induction.