Behaviour of x[n] = cαⁿ

With $c$ and $\alpha$ both real, $x[n] = c\alpha^n$ is a real exponential sequence. Its shape depends on the value of $\alpha$ (take $c > 0$):

For negative $\alpha$, $\alpha^n$ is positive for even $n$ and negative for odd $n$, giving the alternating pattern. A negative $c$ simply flips the whole sequence.

 0 < alpha < 1: decays
  o
  |
  |  o
  |  |  o  o
--+--+--+--+--o-> n
  0  1  2  3  4

 alpha > 1: grows
              o
              |
           o  |
        o  |  |
  o  o  |  |  |
--+--+--+--+--+-> n
  0  1  2  3  4

 -1 < alpha < 0: alternates, decays
  o
  |
  |
  |     o
--+--+--+--+--o-> n
     |     o
     o
  0  1  2  3  4

 alpha < -1: alternates, grows
        o
        |
  o     |
--+--+--+--+-> n
     |     |
     o     |
           |
           o
  0  1  2  3

Examples: $(0.5)^n u[n]$ decays (used for stable system impulse responses); $(-0.8)^n u[n]$ oscillates and decays; $2^n u[n]$ grows (unstable).