You definitely want to convert to polar form, rcisΘ, which is shorthand for r⋅(cosΘ + isinΘ). rcosΘ represents the pure real component (graphed on the real, x, axis) while (rsinΘ)i is the pure imaginary component of the complex number. Graphing 1 - i in the complex plane shows r = √2 (distance from origin, √a2+b2) and Θ = -π/4 (the angle whose terminal ray passes through the pt, therefore, Θ =tan-1(b/a)).
All of this makes multiplication and exponentiation of complex numbers much easier:
1 + √2cis-π/4 + 2cis-π/2 + 2√2cis-3π/4 + 4cis-π + ... + 16 + 16√2cis-π/4
The 2nd to last term above is (√2)8cis-2π which = 16cis0 = 16. In the last term above, -9π/4 is coterminal with -π/4. Converting the 10 terms back into rectangular form will result in a lot of simplification, and a final answer that looks like (# + #√2) + (# + #√2)i, I'm guessing.
