Is there a rule for evaluating a sum like $\sum_{x=1}^{10} (x + c)^3$ where $c$ is a constant?

Question

I'm kind of new to summations and wonder if there is a rule to sum cubic terms. For example $$\sum_{x=1}^{10} (x + c)^3$$ where $c$ is a constant.

I know how to calculate this sum, but I wonder if there is a rule such as distributing the summation inside the parenthesis?

Answer 1

There's a common formula for sums of cubes: $\sum_{x = 1}^n x^3 = \frac 1 4n^2(n+1)^2$. So we have \begin{align*} \sum_{x = 1}^n (x + c)^3 &= \sum_{x = c + 1}^{n+c} x^3 \\ & = \sum_{x = 1}^{n+c} x^3 - \sum_{x = 1}^{c} x^3 \\ & = \frac 1 4(n+c)^2(n+c+1)^2 - \frac 1 4 c^2(c+1)^2. \end{align*}

I know how to calculate this sum but wonder if there is a rule such as distributing the summation inside the parenthesis?

I think you can simplify the above expression a little bit further but in general sums and powers can't be easily interchanged.

Answer 2

Expand ${(x+c)}^3$ into a polynomial then you can apply the summation to each term:

$$\sum_{x=1}^{10} x^3 + \sum_{x=1}^{10} 3x^2c + \sum_{x=1}^{10} 3xc^2 + \sum_{x=1}^{10} c^3$$