Question

The explicit rule for a sequence is given. An = 4n - 1

Select the recursive rule for the geometric sequence?
A) a_1 = 4; a_n = ja_n-1
B) a_n = -4a_n-1
C) a_1 = 0; a_n = 4a_n-1
D) a_n = 4a_n-1.

Answer 1

The explicit sequence An = 4n - 1 does not represent a geometric sequence, therefore, a recursive rule with a common ratio for a geometric sequence cannot be derived from it.

Step-by-step explanation:

The explicit rule for a sequence is An = 4n - 1. To find a corresponding recursive rule for a geometric sequence, we need to recognize the relationship between successive terms. Given that a geometric sequence has a common ratio, we look for a multiplication factor from one term to the next.

Examining the explicit rule, we see that each term is four times the term number minus one. This doesn't directly reveal a multiplication pattern between terms. However, we can calculate the first few terms to unveil any potential patterns:

• a1 = 4(1) - 1 = 3
• a2 = 4(2) - 1 = 7
• a3 = 4(3) - 1 = 11

Since these values do not result in a common multiplication ratio, we can conclude that we are not dealing with a geometric sequence, and thus a recursive rule of the form specified in the options does not exist.