Question

Given the sequence aₙ = 12, 4, 4 * 3, 4 * 9, ..., what is the explicit formula for aₙ?

A. aₙ = 12 * 3^(n-1)
B. aₙ = 12 * 4^(n-1)
C. aₙ = 12 * 9^(n-1)
D. aₙ = 12 * 3^n

Answer 1

The explicit formula for the given sequence is aₙ = 12 × 3^(n-1), which is represented by option A. The sequence shows that starting from the second term, each term is 4 multiplied by a power of 3, requiring the adjustment for the first term which is 12.

Step-by-step explanation:

The sequence given is a₁ = 12, a₂ = 4, a₃ = 4 × 3, a₄ = 4 × 9, ..., and we need to find the explicit formula for aₙ. On closer analysis, we see that from the second term onward, each term is a product of 4 and some power of 3: a₂ = 4 (which is 4 × 3⁰), a₃ = 4 × 3 (which is 4 × 3¹), a₄ = 4 × 9 (which is 4 × 3²), and so on.

We can thus express the nth term of the sequence with n greater than 1 as follows: aₙ = 4 × 3(n - 2). Remembering that the first term a₁ is 12, we can adjust our formula to include the first term by rewriting it as aₙ = 12 × 3(n - 1).

The correct option which represents this explicit formula is answer choice A: aₙ = 12 × 3(n - 1). Therefore, the student should select option A as the correct answer that gives the explicit formula for the sequence aₙ.