Question

Bridgette is organizing her books and putting them on shelves. She put 49 books on the first shelf, 65 books on the second shelf, 83 books on the third shelf, and 103 books on the fourth shelf. What kind of sequence is this?

Answer 1

is it arithmetic, geometric, both, neither

Answer 2

To determine the kind of sequence that Bridgette's books on the shelves form, we can analyze the number of books on each shelf:

- First shelf: 49 books
- Second shelf: 65 books
- Third shelf: 83 books
- Fourth shelf: 103 books

Next, we find the differences between the number of books on successive shelves:

1. Difference between the second and first shelf: \(65 - 49 = 16\)
2. Difference between the third and second shelf: \(83 - 65 = 18\)
3. Difference between the fourth and third shelf: \(103 - 83 = 20\)

Now, the differences are:

- First difference: 16
- Second difference: 18
- Third difference: 20

The pattern of differences shows that they are increasing by 2 each time:

- The second difference (18) is 2 more than the first difference (16).
- The third difference (20) is 2 more than the second difference (18).

Since the first differences are not constant but show a consistent increase of 2, this sequence can be classified as a quadratic sequence. A quadratic sequence is characterized by having constant second differences, indicating that the sequence can be described by a quadratic polynomial. Thus, Bridgette's books follow a quadratic sequence.

Answer 3

To classify the sequence of book counts as arithmetic, geometric, both, or neither, let's review the definitions:

1. **Arithmetic Sequence**: A sequence where the difference between consecutive terms is constant.
2. **Geometric Sequence**: A sequence where the ratio between consecutive terms is constant.

Let's look at the sequence again:

- First shelf: 49 books
- Second shelf: 65 books
- Third shelf: 83 books
- Fourth shelf: 103 books

**1. Checking for Arithmetic Sequence:**
- Differences:
- \(65 - 49 = 16\)
- \(83 - 65 = 18\)
- \(103 - 83 = 20\)

The differences (16, 18, 20) are not constant, which means this is **not an arithmetic sequence**.

**2. Checking for Geometric Sequence:**
- Ratios:
- \( \frac{65}{49} \approx 1.3265\)
- \( \frac{83}{65} \approx 1.2769\)
- \( \frac{103}{83} \approx 1.2398\)

The ratios are also not constant, meaning this is **not a geometric sequence**.

Since the sequence does not meet the criteria for either an arithmetic or a geometric sequence, the answer is **neither**.

Therefore, the sequence of books Bridgette put on her shelves is neither arithmetic nor geometric. It is actually a quadratic sequence as established earlier due to the changing first differences.