Affine combination absolute sum?

Question

For an equation $\sum_{k=0}^n c_k x^k$, i have coefficients which have the affine combination property $\sum_{k=0}^n c_k=1$. Upon taking the absolute sum, i found that i get $\sum_{k=0}^n |c_k|=n$. I know that by the triangle inequality $|\sum_{k=0}^n c_k | \leq \sum_{k=0}^n |c_k|$ which seemingly makes this a special case, but i haven't read in a book or paper that the absolute sum is necessarily equal to $n$? If anyone knows why this is, or if the equation has to have some sort of property for this to happen, or is this a special case of affine combination coefficients?

Answer 1

$$\sum_{k=0}^n|c_k|=\sum_{k=0}^{n-1}|c_k|+\left|1-\sum_{k=0}^{n-1}c_k\right|$$ where the $n-1$ $c_k$ are unconstrained. This expression can take arbitrary values.