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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?

Verified Answer :

Nirina7

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:

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