<s>[INST] Sum the first $n$ terms of the series $1 \cdot 3 \cdot 2^2 + 2 \cdot 4 \cdot 3^2 + 3 \cdot 5 \cdot 4^2 + \cdots$ The question
Sum the first $n$ terms of the series:
$$ 1 \cdot 3 \cdot 2^2 + 2 \cdot 4 \cdot 3^2 + 3 \cdot 5 \cdot 4^2 + \cdots. $$
This was asked under the heading using method of difference and the answer given was
$$ S_n = \frac{1}{10}n(n+1)(n+2)(n+3)(2n+3). $$
My approach
First, I get $$ U_n=n(n+2)(n+1)^2. $$
Then I tried to make $U_n = V_n - V_{n-1}$
in order to get $S_n = V_n - V_0$.
But I really don't know how can I figure this out.
 [/INST] Rewrite the formula as
$$\sum_2^{n+1}k^2(k+1)(k-1)=\sum_2^{n+1}k^4-\sum_2^{n+1}k^2.$$
It is a well known formula that
$$1+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}.$$
Due to Fermat, we also have
$$1+2^4+3^4+\cdots+n^4=\frac15n^5+\frac12n^4+\frac13n^3-\frac1{30}n.$$
Just substitute in your values to get the result.
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