Adele went out to her garden and cut 4 roses from the first rose bush, 9 roses from the second rose bush, 16 roses from the third rose bush, and 25 roses from the fourth rose bush. What kind of sequence is this

Answer 1

The number of roses cut from each rose bush is as follows: 4, 9, 16, and 25.

To determine the type of sequence, we can examine the numbers:

• 4 is \(2^2\)
• 9 is \(3^2\)
• 16 is \(4^2\)
• 25 is \(5^2\)

These numbers are perfect squares of consecutive integers: \(2^2\), \(3^2\), \(4^2\), and \(5^2\).

Thus, the sequence represents the squares of the integers starting from 2. Therefore, it can be described as a sequence of perfect squares (specifically, the squares of the integers starting from 2) or more generally, a quadratic sequence.

Answer 2

is it arithmetic, geometric, both, neither

Answer 3

To determine if the sequence \(4, 9, 16, 25\) is arithmetic, geometric, both, or neither, we can analyze the characteristics of each type of sequence:

1. Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant.

   • The differences are:
     • \(9 - 4 = 5\)
     • \(16 - 9 = 7\)
     • \(25 - 16 = 9\)
   • The differences are not constant (5, 7, 9), so it is not arithmetic.

2. Geometric Sequence: In a geometric sequence, the ratio between consecutive terms is constant.

   • The ratios are:
     • \( \frac{9}{4} = 2.25 \)
     • \( \frac{16}{9} \approx 1.78 \)
     • \( \frac{25}{16} \approx 1.56 \)
   • The ratios are also not constant, so it is not geometric.

Since the sequence is neither arithmetic nor geometric, the answer is that it is neither.