# Natalia is writing a recursive formula to represent the sequence. 8, 12, 18, 27, ... What value should she use as the common ratio in the formula?

Natalia is writing a recursive formula to represent the sequence. 8, 12, 18, 27, … What value should she use as the common ratio in the formula?

Natalia is writing a recursive formula to represent the sequence. 8, 12, 18, 27, … What value should she use as the common ratio in the formula?

Answer: The sequence 8, 12, 18, 27, … is an arithmetic sequence, meaning that each term is obtained by adding a fixed constant difference to the previous term. To find the common difference, we can subtract any two consecutive terms. For example:

12 - 8 = 4
18 - 12 = 6
27 - 18 = 9

We see that the common difference is increasing, so this is not a geometric sequence. Instead, we can use the formula for the nth term of an arithmetic sequence:

an = a1 + (n - 1)d

where an is the nth term, a1 is the first term, n is the number of the term, and d is the common difference.

For this sequence, a1 = 8 and d = 4, so the formula for the nth term is:

an = 8 + (n - 1)4

We can simplify this formula to:

an = 4n + 4

If Natalia wants to write a recursive formula for this sequence, she would need to write the formula for the nth term in terms of the previous term. Since this is not a geometric sequence, we cannot use a formula of the form an = r * an-1. Instead, we can use the formula for the difference between consecutive terms:

an - an-1 = d

Substituting the formula for an, we get:

(4n + 4) - (4(n - 1) + 4) = 4

Simplifying this equation, we get:

an = an-1 + 4

Therefore, the recursive formula for this sequence is:

an = an-1 + 4

where a1 = 8. This formula says that each term is obtained by adding 4 to the previous term.
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