Question # Determine whether the given sequence could be geometric, arithmetic, or neither. If possible, identify the common ratio or difference. 9,\ 13,\ 17,\ 21,\ \cdots

Polynomial arithmetic
ANSWERED Determine whether the given sequence could be geometric, arithmetic, or neither.
If possible, identify the common ratio or difference.
$$\displaystyle{9},\ {13},\ {17},\ {21},\ \cdots$$
a) arithmetic: $$\displaystyle{d}={4}$$
b) arithmetic: $$\displaystyle{r}={4}$$
c) arithmetic: $$\displaystyle{r}={\frac{{{1}}}{{{4}}}}$$
d) neither 2021-08-07
Step 1
For a sequence to be arithmetic sequence difference between consecutive terms must be equal.
Mathematically,
$$\displaystyle{d}={t}_{{{2}}}-{t}_{{{1}}}={t}_{{{3}}}-{t}_{{{2}}}$$
Consider the given secuence:
$$\displaystyle{9},\ {13},\ {17},\ {21},\ \cdots$$
Step 2
Now check for common difference,
$$\displaystyle{d}={13}-{9}={4}$$
$$\displaystyle{d}={17}-{13}={4}$$
This proves that,
$$\displaystyle{4}={t}_{{{2}}}-{t}_{{{1}}}={t}_{{{3}}}-{t}_{{{2}}}$$
Therefore, it is a arithmetic progression with common difference 4.