Answer:
Z_1 = 3
Z_2 = 2.5
Z_3 = 2.25
Explanation:
To find the next three terms in the sequence defined by Z_0 = 4 and Z_n = (1/2) Z_(n-1) + 1, we can use the recursive formula to calculate each term.
First, we can calculate Z_1 by substituting n = 1 into the recursive formula:
Z_1 = (1/2) Z_0 + 1 = (1/2) (4) + 1 = 3
This means that the second term in the sequence is 3.
Next, we can calculate Z_2 by substituting n = 2 into the recursive formula:
Z_2 = (1/2) Z_1 + 1 = (1/2) (3) + 1 = 1.5 + 1 = 2.5
This means that the third term in the sequence is 2.5.
Finally, we can calculate Z_3 by substituting n = 3 into the recursive formula:
Z_3 = (1/2) Z_2 + 1 = (1/2) (2.5) + 1 = 1.25 + 1 = 2.25
This means that the fourth term in the sequence is 2.25.
Therefore, the next three terms in the sequence are 3, 2.5, and 2.25.
Each term in the sequence is calculated by taking half of the previous term and adding 1. The sequence starts at 4, so the second term is calculated by taking half of 4, which is 2, and adding 1, which gives 3. The third term is calculated by taking half of 3, which is 1.5, and adding 1, which gives 2.5. The fourth term is calculated by taking half of 2.5, which is 1.25, and adding 1, which gives 2.25. This pattern continues for each subsequent term in the sequence.