Sum of 12 numbers

Question

Which formula can be used to calculate the sum:

$$\sum\limits_{i=1}^{12} \left(1 + \frac{i^2 -i}{2}\right)~~?$$

Is there any formula that can be applied instead of manually adding the 12 numbers?

Answer 1

Notice that with $S(n):=(n+1)n(n-1)$ we have

$$S(n)-S(n-1)=(n+1)n(n-1)-n(n-1)(n-2)=3n(n-1).$$

So by telescoping, the sum of $i^2-i$ from $1$ to $12$ is

$$\frac{S(12)-S(0)}3.$$

Hence

$$\frac{13\cdot12\cdot11}{3\cdot2}+12.$$

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Note that this "trick" generalizes to more factors:

$$(n+1)n(n-1)\cdots(n-k+1)-n(n-1)(n-2)\cdots(n-k) \\=(k+1)n(n-1)\cdots(n-k+1).$$