What type of sequences would these be?

Question

So as far as I can tell, there are two types of possible sequences:

1. Arithmetic sequences $\to$ Where there's a linear/quadratic/cubic/... progression i.e. with a formula of $c_yn^y+c_{y-1}n^{y-1}+c_{y-2}n^{y-2}+\dots+c_0$
2. Geometric sequences $\to$ Where the ratio between a term and that before it is constant i.e. with a formula of $a^n$

I'm not sure if I'm correct so far, so please correct me if I'm not!

As we can see, geometric sequences are primarily defined by calculating the exponent of a certain number. But what if you added several of these together?

So, if a formula for a sequence was $2^n + 3^n - 4^n + 5^n$, what type of sequence would it be? Also, how would it be solved if you were just given the terms?

Taking this one step further, what if the formula for the sequence was $2n^2 - 3n + 5^n - 4^n + 5\cdot(2^n)$? How would this be solved?

Answer 1

I feel like I need to expand on the comments of @DonAntonio and @David K.

First of all, an arithmetic sequence is a sequence where the difference of any $2$ consecutive terms is constant (notice the similarity with the geometric one). Notice that there are infinitely many sequences besides the $2$ types you listed- take for example

$$1,0,0,0,0,0,0\dots$$

defined by $a_1=1, a_n=0$ for $n\geq2$. This is neither an arithemetic progression nor a geometric one.

In particular, a sequence is a function $f:\mathbb{N}\rightarrow S$ where $S$ is an object and the sequence $(a_n)_{n\in\mathbb{N}}$ is defined as

$$a_i=f(i)$$

Since you can make the function arbitrary, there are a lot of sequences not satisfying neither of the $2$ properties you stated.

Also, you can see that even if you have a finite number of terms, then unless you are told something about the general form of the sequence, you are not able to find a unique sequence which starts the same as the finitely many terms you have since again, you can set the next entry to be an arbitrary one.

Having said all of this, informally you can think of a sequence as a list of objects (numbers in your case). Since you can list your objects however you like, you see that there is no general rule for determining a sequence.