Question

Given the recursive definition:

n = 1, eh_n = 12
n > 1, eh_n = (1/3) * eh_(n-1)

What is the explicit definition for the sequence?

A. eh_n = 12^n
B. eh_n = 12 + 1/3^(n-1)
C. eh_n = 12 + (1/3)^n
D. eh_n = 12 * (1/3)^(n-1)

Answer 1

The explicit definition for the sequence is ehn = 12 * (1/3)(n-1).

Step-by-step explanation:

The recursive definition given is:

n = 1,

ehn = 12 for n > 1,

ehn = (1/3) * eh(n-1)

To find the explicit definition for the sequence, we need to express ehn in terms of n. Let's consider the first few terms of the sequence:

eh1 = 12

eh2 = (1/3) * eh1 = (1/3) * 12 = 4

eh3 = (1/3) * eh2 = (1/3) * 4 = 4/3

From the pattern, we can observe that ehn is decreasing by a factor of 1/3 for each subsequent term. Therefore, the explicit definition for the sequence is ehn = 12 * (1/3)(n-1). Answer choice D is the correct answer.