Let f(z) be the degree n polynomial z^n + 2*z^(n-1) + 3*z^(n-2) + ... + (n-1)*z^2 + n*z + n+1.

Prove all n roots of f(z) have a magnitude greater than 1; i.e. if f(z)=0 then |z|>1.

A

pentagram is typically constructed by taking the diagonals of a regular pentagon.

This common pentagram has five angles measured at the vertices, each of which equals 36 degrees. In total all five angles sum to 180 degrees.

Generalize to make an irregular pentagram by taking the diagonals of a convex irregular pentagon.

This irregular pentagram also has five angles measured at the five vertices. Show that the sum of these five angles equals 180 degrees.