Showing that progression is not arithmetic or geometric

Question

The progression $(w_n)$ is defined by:

$$w_ n =2^n - 2n + 2 $$

I must show that this progression is not arithmetic and also that it is not geometric. I know how to show that a progression is arithmetic or geometric but I never worked with this type of exercise.

What shall I do?

Thanks for your answers.

Answer 1

Reason by contradiction. If a sequence were arithmetic, then consecutive terms would have a common difference $d$. If a sequence were geometric, then consecutive terms would have a common ratio $r$. Show that no such $d$ nor $r$ exist.

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I should also add that it is unnecessary to do the computation for general $n$: it suffices to find particular terms of the sequence that fail to have such $d$ or $r$. For example, if you simply compute $w_1, w_2, w_3$, you can tell right away that the sequence is neither arithmetic nor geometric.

Answer 2

$$w_ n =2^n-2n+2$$ $$w_ {n+1} =2^{n+1}-2(n+1)+2$$ $$w_ {n+2} =2^{n+2}-2(n+2)+2$$

1. If $$w_n-w_{n-1}=w_{n-1}-w_{n-2}$$ you have arithmetic series

2. If $$\frac{w_n}{w_{n-1}}=\frac{w_{n-1}}{w_{n-2}}$$ you have geometric series