Question

The nth term of the sequence

-2, 1, 4, 7, 10,...

is given by the formula
A) an = n - 3
B) an = 3n - 3
C) an = n - 5
D) an = 3n - 5

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= (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= (10)/(7)`
- For
`a_(n)=4` and
`a_(n-1)=1`:

`d=4-1`

`d=3`

`r= (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