Question

Use iteration to guess an explicit formula for the sequence tₖ defined by the recurrence relation tₖ = tₖ₋₁+3k, for every integer k≥1. Which of the following explicit formulas represents the sequence?

Answer 1

To guess an explicit formula for the sequence tₖ defined by the recurrence relation tₖ = tₖ₋₁ + 3k, we can use iteration to find a pattern in the sequence. The explicit formula for the sequence is tₖ = 3k² + 3k.

Step-by-step explanation:

To guess an explicit formula for the sequence tₖ defined by the recurrence relation tₖ = tₖ₋₁ + 3k, we can use iteration to find a pattern in the sequence.

Let's start with the initial condition t₁ = 3 + 3(1) = 6. This gives us the first term of the sequence.

Then, we can use the recurrence relation to find the next term: t₂ = t₁ + 3(2) = 6 + 6 = 12. Continuing this process, we can find the values of t₃, t₄, and so on. By examining these values, we can guess a formula for the sequence.

After calculating several terms, we can observe that the sequence tₖ can be represented by the explicit formula tₖ = 3k² + 3k.