Question

In exploration 1.4.2, you explored the differences involving sequences in the form _____. Which of the following is the strongest conjecture for a sequence with the kth power?

a. Linear sequences
b. Geometric sequences
c. Exponential sequences
d. Quadratic sequences

Answer 1

An exponential sequence is the strongest conjecture for a sequence with the kth power due to the characteristic increase by a constant factor raised to the position's power in the sequence.

Step-by-step explanation:

The strongest conjecture for a sequence with the kth power is c. Exponential sequences. This is because exponential sequences are defined by their terms increasing by a constant factor raised to the power of their position in the sequence. Taking the doubling sequence as an example, it starts at 1 and then each term is 2 raised to the power of its position: 21, 22, 23, and so on, with each term being twice the previous one. Therefore, after n intervals, the increase is by a factor of 2n.

The strongest conjecture for a sequence with the kth power is exponential sequences. In an exponential sequence, each term is found by multiplying the previous term by a constant number, known as the base. For example, the sequence 2^0, 2^1, 2^2, 2^3, 2^4 follows an exponential pattern with a base of 2. Exponential sequences are characterized by exponential growth or decay.