Find the difference equation for {2, 4, 16, 256, ...}

Question

Write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence.

b. {2,4,16, 256,...}

I know that a_n= 2²ⁿ but I can't figure out how to find Δa_n = c*a_n.

I'm lost. I'd really appreciate some help.

Answer 1

The ratio $a_n/a_{n-1}$ is always $a_{n-1}$. So the term difference is $$ \Delta a_n \equiv a_n-a_{n-1} = a_{n-1}^2 - a_{n-1} = ( a_{n-1} -1) a_{n-1} $$ If you insist on writing this as $c \cdot a_n$ the form would be $$ \Delta a_n = \frac{a_{n-1}-1}{a_{n-1}} a_n $$