Question

A certain arithmetic sequence has the following explicit formula for the nth term:

an = 2 + (n - 1)(6)

The same sequence has the following recursive formula:

an = an-1 + _____

What number belongs in the blank space in the recursive formula?

Answer 1

arithmetic sequence general term is an+1 - an =an - an-1= ... = a2 -a1=c
but we have an = 2 + (n - 1)(6) as an expression of the sequence, so c=6
and then an - an -1 =c = 6, from where an = an-1 + 6

Answer 2

The nth term for the arithmetic sequence is given by:

`a_n = a_1+(n-1)d`

where,

`a_1` is the first term

n is the number of terms and

d is the common difference for two consecutive terms.

As per the statement:

A certain arithmetic sequence has the following explicit formula for the nth term:

`a_n = 2+(n-1)(6)`

⇒
`a_1 = 2` and d = 6

The recursive formula for the arithmetic sequence is given by:

`a_n = a_(n-1)+d`

Substitute the given values we have;

`a_n = a_(n-1)+6`

⇒the following recursive formula is,
`a_n = a_(n-1)+6`

Therefore, the number belongs in the blank space in the recursive formula is, 6