<s>[INST] How to compute $1 \times 2 \times 3 \times 4 + 3 \times 4 \times 5 \times 6 + ... + 97 \times 98 \times 99 \times 100$ How to compute $S = 1 \times 2 \times 3 \times 4 + 3 \times 4 \times 5 \times 6 + ... + 97 \times 98 \times 99 \times 100$
Was thinking $\frac{S}{24} = {4\choose 4} + {6\choose 4} + {8\choose 4} + ... + {100\choose 4}$ but how do I sum this up?
 [/INST] General term is:
$$n(n+1)(n+2)(n+3)=n^4+6n^3+11n^2+6n$$
Now we give values 1, 3,... n :
$1\times2\times3\times4=1^4+6\times1^3+11\times 1^2+6\times 1$
$3\times4\times5\times6=3^4+6\times3^3+11\times 3^2+6\times 3$
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$n(n+1)(n+2)(n+3)=n^4+6n^3+11n^2+6n$
Now we sum up:
$A=S_4+6S_3+11S_2+6S_1$
We use  well known formulas for $S_n$ for odd numbers, for example:
$S_3=1^3+3^3+ . . . +n^3=n^2(2n^2-1)$
Here $n= 96$, we plug this in and get the sum.
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