Final answer:
A geometric sequence with a starting term of 12 and a common ratio of ⅓ can be recursively defined by a1 = 12 and an = ⅓an-1. The graph of the first five terms shows a rapid decline toward zero.
Step-by-step explanation:
The geometric sequence provided starts with 12, 6, ... To write a recursive definition for this sequence, we note that each term is obtained by multiplying the previous term by a constant ratio. Here, the second term is half the first term (6 is half of 12), so the ratio is 0.5 or ⅓. A recursive definition for the sequence is: a1 = 12 and an = ⅓an-1 for n ≥ 2.
To sketch a graph representing the first 5 terms of the sequence, you would plot the term number on the x-axis and the term value on the y-axis. The first five terms would be 12, 6, 3, 1.5, and 0.75, creating a graph that rapidly approaches zero as the term number increases.