I have this function: $y = x^3$
I'm trying to obtain the sum of all the $y$ values for every integer $x$ for example in the range $[1, \dots, 10]$. I know that for numerical progression (for example $[1,..,10]$) we can do:
$(f/2)*(i+f)$
Where $i$ is the initial value $(1)$ and $f$ is the final value $(10)$. And we'll obtain the equivalent of $1+2+3+4+5+6+7+8+9+10$.
Is there an equation to obtain the same result as the operation $(1 + (2^3) + (3^3) + \dots )$
I've looked for an aswer online but I could not find anything. Is there an actual way of doing it?
Let's say there is a function
$$S(x) = \sum_{i = 0}^{x} i^3$$
Now, this could be a polynomial with unknown parameters, for example
$$S(x) = a x^4 + b x^3 + c x^2 + d x + e$$
Now, say this function is the solution and satisfies
$$S(0) = 0$$
$$S(1) = 1$$
$$S(2) = 1 + 2^3$$
$$S(3) = 1 + 2^3 + 3^3$$
$$S(4) = 1 + 2^3 + 3^3 + 4^3$$
Now you have a system of 5 linear equations (which immediately reduces to four, because $e = 0$), which upon solving should give you the desired parameters for $S(x)$.
You can look a solution up here: https://brilliant.org/wiki/sum-of-n-n2-or-n3/