Recursive Sequences

Front

What is a recursive sequence?

Back

A recursive sequence is a sequence of numbers where each term is defined as a function of one or more of the preceding terms.

Front

What is the general form of a recursive formula?

Back

A recursive formula typically has the form a_n = f(a_(n-1), a_(n-2), ..., n), where f is a function that defines the relationship between terms.

Front

What is the first term in the sequence defined by a_n = 2a_(n-1) + n with a_0 = 0?

Back

The first term a_1 = 1.

Front

How do you find the next term in a recursive sequence?

Back

To find the next term, apply the recursive formula using the previous term(s) in the sequence.

Front

What is the recursive rule for the sequence {16, 8, 4, 2, ...}?

Back

The recursive rule is a_n = a_(n-1) * (1/2) with a_1 = 16.

Front

What is the rebound ratio in the context of a bouncing ball?

Back

The rebound ratio is the fraction of the height to which a ball rebounds after hitting the ground.

Front

What is the recursive rule for the Fibonacci sequence?

Back

The recursive rule is f_n = f_(n-1) + f_(n-2) with initial conditions f_0 = 0 and f_1 = 1.

Front

How do you determine if a sequence is geometric?

Back

A sequence is geometric if each term after the first is found by multiplying the previous term by a constant called the common ratio.

Front

What is the common ratio in the sequence {7, 35, 175, 875, ...}?

Back

The common ratio is 5, as each term is multiplied by 5 to get the next term.

Front

What is the difference between a recursive sequence and an explicit formula?

Back

A recursive sequence defines each term based on previous terms, while an explicit formula allows you to calculate any term directly without referring to previous terms.