I am currently doing a Financial Mathematics module and am currently rather stumped by a question involving Itô's Lemma and Riemann sums. The question goes as follows:

We know Itô's Lemma implies that

$$3\int_{0}^{T} (W_t)^2 dW_t = W_T^3 - W_0^3 - 3\int_{0}^{T} W_t dt$$ Show that the result can also be found by writing the integral as $$3\int_{0}^{T} W_t^2 dW_t = \lim_{N\to\infty} 3\sum_{i=0}^{N-1} W_i^2(W_{i+1} -W_i)$$

I know that from a function $F(W_T,T) = W_T^3$ we can write Itô's Lemma as above (the first equation) and I also know from some lectures that when $G(W_T,T) = W_T^2$ we can write

$$2\int_{0}^{T} W_t dW_t = \lim_{N\to\infty} 2\sum_{i=0}^{N-1} W_i(W_{i+1} - W_i)$$

and then use a nice trick that $$2W_{i}(W_{i+1} - W_i) = W_{i+1}^2 - W_i^2 - (W_{i+1} - W_i)^2$$

so we can simplify the Itô Stochastic Integral to $$ = W_T^2 - W_0^2 -T$$

I have a feeling I need to find a similar trick so I can simplify $W_i^2(W_{i+1} - W_i)$ and thus simplify the summation and end up at the result but I am really struggling to find the right simplification required. Any hints would be greatly appreciated.

Note: I would much prefer hints so I can reach the solution of my own accord than someone simply providing a solution.


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