<s>[INST] Sum of the series $1\cdot3\cdot2^2+2\cdot4\cdot3^2+3\cdot5\cdot4^2+\cdots$ Find the sum of $n$ terms of following series:

$$1\cdot3\cdot2^2+2\cdot4\cdot3^2+3\cdot5\cdot4^2+\cdots$$

I was trying to use $S_n=\sum T_n$, but while writing $T_n$ I get a term having $n^4$ and I don't know $\sum n^4$. Is there any other way find the sum?
 [/INST] An alternative to finding the formula for $\sum k^4$ is to instead write the $k$th term as $$(k+2)(k+1)^2k=(k+3)(k+2)(k+1) k-2(k+2)(k+1)k.$$
That is, one trades sums of integer powers for sums of consecutive products. To see the advantage, note for example that
$$1\cdot 2\cdot 3 +2\cdot 3\cdot 4+3\cdot 4\cdot 5+4\cdot 5\cdot 6=210= \frac{1}{4}(4\cdot 5\cdot 6\cdot 7),$$
i.e. the sum of products of three consecutive integers can be expressed in terms of a product of four consecutive integers. A similar pattern holds for other such sums, allowing the required formulas to be readily found. (By comparison, the pattern for sums of integer powers is much more involved.) From such computations one can then obtain the desired formula for the original sum.
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