Question

What is the recursive formula for the geometric sequence with this explicit formula?

(a) a_(n+1) = -8 * a_n
(b) a_(n+1) = 5 * a_(n-1)
(c) a_(n+1) = 0.5 * a_n
(d) a_(n+1) = 2 * a_n

Answer 1

The recursive formula for a geometric sequence reflects the relationship between successive terms along with the initial condition(s). In the case of these options, they are simply the provided multiplicative relations of each term with the previous, with the necessary initial condition.

Step-by-step explanation:

The recursive formula for a geometric sequence can be determined by looking at the relationship between successive terms. Here, we are given the explicit formula in four different versions:

• a_(n+1) = -8 * a_n
• a_(n+1) = 5 * a_(n-1)
• a_(n+1) = 0.5 * a_n
• a_(n+1) = 2 * a_n

For each of these, the recursive formula is simply the given relation, accompanied by the initial condition. For example, for (a), the recursive formula is a_n = -8 * a_(n-1), given a_1 or a_0, the first term. Similarly, for (b), we adjust the index to get a_n = 5 * a_(n-2) with two initial conditions, a_1 and a_2, provided.