Question

Identify a possible explicit rule for the nth term of the sequence 2, 8, 26, 80, 242.

Answer 1

a₁ = 1

a₂ = 5

a₃ = 17

a₄ = 53

a₅ = 161

Explanation:

To find the first five terms of the sequence, we can apply the given recursive formula:

a₁ = 1

For n ≥ 2:

aₙ = 3aₙ₋₁ + 2

Let's calculate the first five terms step by step:

a₁ = 1

For n = 2:

a₂ = 3a₁ + 2 = 3(1) + 2 = 5

For n = 3:

a₃ = 3a₂ + 2 = 3(5) + 2 = 17

For n = 4:

a₄ = 3a₃ + 2 = 3(17) + 2 = 53

For n = 5:

a₅ = 3a₄ + 2 = 3(53) + 2 = 161

Therefore, the first five terms of the sequence are:

a₁ = 1

a₂ = 5

a₃ = 17

a₄ = 53

a₅ = 161