Let a₀, a₂, . . . , a_n be real numbers satisfying

a₀/1 + a₁/2 + a₂/3 + · · · + a_n/(n + 1) = 0.

Show that the equation

a₀ + a₁x + a₂x² + · · · + a_nxⁿ = 0

has at least one real root.