Question

The nth term of the sequence

Answer 1

First we need to find what kind of sequence we have. To do that we are going to test if the sequence as a difference $d$, in which case it will be an arithmetic sequence, or a ratio $r$, in which case it will be geometric sequence.
The formula to find $d$ is $d=a_{n}-a_{n-a}$
where 
$a_{n}$ is the current term in the sequence
$a_{n-1}$ is the previous term in the sequence 
The formula to find $r$ is $r= \frac{a _{n} }{a _{n-1} }$
where 
$a_{n}$ is the current term in the sequence 
$a_{n-1}$ is the previous term in the sequence 
- For $a_{n}=10$ and $a_{n-1}=7$
  $d=10-3$
  $d=3$
  $r= \frac{10}{7}$
- For $a_{n}=4$ and $a_{n-1}=1$:
  $d=4-1$
  $d=3$
  $r= \frac{4}{1}$
  $r=4$
Since $d$ is equal in both procedures whereas $r$ is not, we can conclude that our sequence is an arithmetic sequence.
Now we are going to use the explicit formula of an arithmetic sequence $a_{n}=a_{1}+(n-1)d$
where 
$a_{n}$ is the nth term
$a_{1}$ is the first term 
$n$ is the position of the term in the sequence 
$d$ is the difference 
We know that $a_{1}=-2$ and $d=3$, so lets replace those values in our formula:
$a_{n}=-2+(n-1)(3)$
$a_{n}=-2+3n-3$
$a_{n}=3n-5$

We can conclude that the correct answer is D) an = 3n - 5