A geometric series with geometric rate

Question

Problem:

A geometric series with geometric rate

1) Series C and S are defined as follows:

                             C=2cos x + 4 cos 2x + 8 cos 3x + ... + 2_n cos nx
                             S=2 sin x + 4 sin 2x + 8 sin 3x + ... +  2_n sin nx

Show that C+iS is a geometric series. Hence show that:

                           S=2sin x-2ⁿ⁺¹sin((n+1)x) +2ⁿ⁺² sin nx /5 - 4 cos x

Find a similar expression for C.