Question

Write a recursive formula and an explicit formula for each sequence. (1/2), (3/2), (9/2), (27/2)

Answer 1

The sequence given follows a pattern in which each term is the previous one multiplied by 3. The recursive formula is an = 3 · an-1, a1 = 1/2, and the explicit formula is an = ½ · 3n-1.

Step-by-step explanation:

The sequence given is (1/2), (3/2), (9/2), (27/2). This sequence can be described by both a recursive formula and an explicit formula, because each term can be found by multiplying the previous term by 3, or by using powers of 3 directly.

Recursive Formula

The recursive formula for the sequence is:

• a1 = 1/2
• an = 3 · an-1, for n ≥ 2

Explicit Formula

The explicit formula, or the nth term of the sequence, is:

• an = ½ · 3n-1

Let's see how the explicit formula applies to the given sequence. For n=1, we have a1 = ½ · 30 = ½. For n=2, a2 = ½ · 31 = ½ · 3 = 3/2. And, it continues similarly for subsequent terms.

Answer 2

The recursive formula is a_n = 3 * a_n-1 and the explicit formula is a_n = (3^n) / 2.

Step-by-step explanation:

To find the recursive formula for the given sequence (1/2), (3/2), (9/2), (27/2), we can observe that each term is obtained by multiplying the previous term by 3. So, the recursive formula is given by:

an = 3 * an-1

Now, let's find the explicit formula. We can see that the numerators of the terms are powers of 3 (30, 31, 32, 33). The denominators are always 2. So, the explicit formula for the sequence is:

an = (3n) / 2