Ava is dividing cherries among some bowls. She put 3 cherries in the first bowl, 5 cherries in the second bowl, 8 cherries in the third bowl, and 13 cherries in the fourth bowl. What kind of sequence is this?

Answer 1

The number of cherries in the bowls is: 3, 5, 8, and 13.

To identify the pattern, let's look at the differences between consecutive terms:

• The difference between the second and first bowl: \(5 - 3 = 2\)
• The difference between the third and second bowl: \(8 - 5 = 3\)
• The difference between the fourth and third bowl: \(13 - 8 = 5\)

Now we can summarize the differences:

• First difference: 2
• Second difference: 3
• Third difference: 5

Next, let's look at the differences of the differences (the second differences):

• The difference between the second and first difference: \(3 - 2 = 1\)
• The difference between the third and second difference: \(5 - 3 = 2\)

The second differences are:

• First second difference: 1
• Second second difference: 2

The differences of the second differences (the third differences) are:

• The difference between the second second difference and the first second difference: \(2 - 1 = 1\)

Since the third differences are constant (equal to 1), this indicates that the original sequence follows a second-order linear recurrence relation.

In this case, we can actually see that the series resembles a Fibonacci-like sequence, where each term after the first two is the sum of the two preceding terms, starting from certain initial values.

Thus, this sequence is a Fibonacci-type sequence specifically starting from 3 and 5. The next terms would continue following this pattern if continued.

In conclusion, the sequence of cherries in the bowls is a Fibonacci-like sequence.

Answer 2

What kind of sequence is this?

1, 8, 64, 512, ...

Answer 3

The sequence given is: 1, 8, 64, 512.

To analyze this sequence, we can observe how each term is related to the others.

• The first term is \(1\), which can be expressed as \(2^0\).
• The second term is \(8\), which is \(2^3\).
• The third term is \(64\), which is \(2^6\).
• The fourth term is \(512\), which is \(2^9\).

It looks like the exponents of 2 are increasing by 3 each time:

• \(0\) (for \(2^0\))
• \(3\) (for \(2^3\))
• \(6\) (for \(2^6\))
• \(9\) (for \(2^9\))

This shows that each term can be expressed in the form: \[ a_n = 2^{3(n-1)} \]

Alternatively, we can see that:

• \(1 = 2^{3 \cdot 0}\)
• \(8 = 2^{3 \cdot 1}\)
• \(64 = 2^{3 \cdot 2}\)
• \(512 = 2^{3 \cdot 3}\)

Thus, the sequence is a geometric sequence where each term is multiplied by a common ratio of \(2^3 = 8\).

In conclusion, the sequence is a geometric sequence with a common ratio of \(8\).

Answer 4

John bagged the plastic bottles after a recycling drive. He placed 55 bottles in the first bag, 110 bottles in the second bag, 220 bottles in the third bag, and 440 bottles in the fourth bag. What kind of sequence is this?

Answer 5

The sequence of the number of bottles in each bag is: 55, 110, 220, and 440.

To analyze this sequence, let's identify how each term relates to the previous ones:

1. First term: \(55\)
2. Second term: \(110\) (which is \(55 \times 2\))
3. Third term: \(220\) (which is \(110 \times 2\))
4. Fourth term: \(440\) (which is \(220 \times 2\))

From this, we can see that each term is being multiplied by 2 to get the next term.

We can write this pattern mathematically as: \[ a_n = 55 \times 2^{(n-1)} \] where \(n\) represents the term number in the sequence.

Thus, the sequence is a geometric sequence with:

• A first term of \(55\)
• A common ratio of \(2\)

In conclusion, the sequence of bottles in the bags is a geometric sequence with a common ratio of \(2\).