Question

Consider the geometric sequence: [4, 8, 16, 32, ...] if [n] is an integer, which of these functions generate the sequence? Choose all answers that apply:

1) a(n) = 4(2)ⁿ for [n ≥ 0]
2) b(n) = 2(2)ⁿ for [n ≥ 1]
3) c(n) = 4ⁿ⁻¹ for [n ≥ 2]
4) d(n) = 4ⁿ⁻² for [n ≥ 3]

Answer 1

Functions a(n) = 4(2)^n for n ≥ 0 and b(n) = 2(2)^n for n ≥ 1 generate the given geometric sequence [4, 8, 16, 32, ...]. The other functions do not match the sequence.

Step-by-step explanation:

The geometric sequence in question is [4, 8, 16, 32, ...], and we need to identify which functions generate this sequence for n being an integer. Let's consider each function.

• a(n) = 4(2)n for n ≥ 0: Substituting n = 0, 1, 2, 3, we get 4, 8, 16, 32, respectively, which corresponds exactly to the given sequence.
• b(n) = 2(2)n for n ≥ 1: Substituting n = 1, 2, 3, 4, we get 4, 8, 16, 32, which also fits the sequence perfectly.
• c(n) = 4n-1 for n ≥ 2: Substituting n = 2, 3, 4, 5, we get 4, 16, 64, 256, which does not match the sequence as it increases by a factor of four.
• d(n) = 4n-2 for n ≥ 3: Substituting n = 3, 4, 5, 6, we get 4, 16, 64, 256, which, again, does not match the initial sequence.

Therefore, the functions that generate the given geometric sequence are a(n) and b(n).