Arithmetic sequence explicit formula.

Question

Which arithmetic sequence explicit formula would yield the following: $1$, $-1$, $1$, $-1$.

Answer 1

None. A sequence $\,\{a_1,a_2,...\}\,$ is arithmetic iff $\,a_{n+1}-a_n=d=$constant, for any $\,n\geq 1\,$.

In this case it doesn't work, yet your sequence is a geometric one, since

$$\frac{a_{n+1}}{a_n}=-1=\,\text{constant}$$

and thus a general formula for the n-th element is $a_n=1\cdot(-1)^{n-1}=(-1)^{n-1}\,$

Answer 2

$a_n = (-1)^{n+1}$ because for odd $n$, you get $1$ and for even $n$, $-1$

for $n \in N$ (I count $N$ as 1, 2,...)

Answer 3

the sequence 1, (-1), (-1)(-1), (-1)(-1)(-1), ... is $$(-1)^n$$ for $n = 0,1,2,..$.