Question

Let aₙ be a sequence of real numbers such that aₙ=3 aₙ-1+2, a₁=2. Compute a₂, a₃, a₄, a₅, then guess a formula for aₙ

Answer 1

To find the next terms in the sequence, we apply the recursive formula and calculate a₂, a₃, a₄, and a₅. The formula for aₙ is aₙ = 2^n + 1.

Step-by-step explanation:

To find the values of a₂, a₃, a₄, and a₅ in the sequence aₙ = 3aₙ₋₁ + 2, where a₁ = 2, we can use the recursive formula to calculate each term:

a₂ = 3a₁ + 2 = 3(2) + 2 = 8

a₃ = 3a₂ + 2 = 3(8) + 2 = 26

a₄ = 3a₃ + 2 = 3(26) + 2 = 80

a₅ = 3a₄ + 2 = 3(80) + 2 = 242

By observing the sequence, we can see that the formula for aₙ is aₙ = 2^n + 1. This means that the nth term in the sequence is equal to 2 raised to the power of n, plus 1.