Induction: Absolute value of complex sequence

[Combinatus]

Homework Statement

Define sequence \left[ {z_n} \right]} _{n=1}^{\infty} of points in the complex plane with z_1 = 1 and z_{n+1} = (4+3i)z_n - 1. Show that |z_n| \ge 4^{n-1} for all n \ge 1.

Homework Equations

The Attempt at a Solution

The base case of induction is true, since |z_1| \ge 4^{1-1} \Leftrightarrow 1 \ge 1.

z_1 = 1
z_2 = 3+3i
z_3 = 2+21i
z_4 = -56+90i
z_5 = -2557-717i
z_6 = -8078-10539i
...

I figured I can find a find some f(n), I'll perhaps be able to show the inductive step. However, I'm getting nowhere on that approach. Perhaps I am not intuitive enough, but the points seem to be all over the complex plane. Thoughts? Guidance? Ideas?!

 

[Dick]

The absolute value |4+3i|=5. Think about that.

 

[Combinatus]

I slept on it, yet no results were yielded. That usually means the problem is impossible, or retardedly simple. I'm pretty sure I know which one of those it is...

|4+3i|=5 would seem very useful if z_{n+1} = (4+3i)z_n, but unfortunately, z_{n+1} = (4+3i)z_n - 1. I don't know what to do with it, since I can't figure out how to treat |(4+3i)z_n - 1|.

 

[Dick]

It's not retardedly simple, but it's not impossible either. There's a thing called the inverse triangle inequality. |a-b|>=||a|-|b||. Can you find a way to use that as well?