 Solution

For a nonzero geometric sequence, substitute $H_n=c{q}^n$ into the linear recurrence and cancel $c{q}^{n-1}$:The characteristic equation of a linear recurrence therefore has rootsThe zero sequence $c=0$ is a degenerate solution for any $q$ for which the displayed expression is defined. A nonzero sequence with $q=0$ cannot satisfy the recurrence at $n=1$, since it would require $0=0+H_0$.