Rachel found some empty bins to put cans into for recycling. She put 461 empty cans in the first bin, 461 empty cans in the second bin, 461 empty cans in the third bin, and 461 empty cans in the fourth bin. What kind of sequence is this?

Answer 1

The number of cans Rachel put into each bin forms a constant sequence, or more specifically, it is an example of a constant sequence. In this case, each term (the number of cans in each bin) is the same: 461. Since there is no change in the value from one bin to the next, all terms are equal, and this results in a constant sequence.

Additionally, it can also be described as an arithmetic sequence where the common difference between terms is zero.

Answer 2

While organizing the magazines at the doctor's office, Sanjay put 47 magazines in the first pile, 47 magazines in the second pile, 47 magazines in the third pile, and 47 magazines in the fourth pile. What kind of sequence is this?

Answer 3

The number of magazines Sanjay put into each pile forms a constant sequence. Specifically, each term in the sequence is the same (47 magazines), meaning there is no variation between the piles.

Like before, you can also describe this as an arithmetic sequence where the common difference between consecutive terms is zero, indicating that each term is equal to the others. In summary, since all the piles contain the same number of magazines, it is a constant sequence.

Answer 4

What kind of sequence is this?

1, 10, 100, 1,000, ...

Answer 5

The sequence you provided—1, 10, 100, 1,000, ...—is a geometric sequence. In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

In this case, the common ratio is 10, since each term is obtained by multiplying the previous term by 10:

• \( 1 \times 10 = 10 \)
• \( 10 \times 10 = 100 \)
• \( 100 \times 10 = 1,000 \)

Thus, this sequence increases exponentially as it progresses.

Answer 6

While sorting some buttons, Gabriel put 1 button in the first box, 5 buttons in the second box, 25 buttons in the third box, and 125 buttons in the fourth box. What kind of sequence is this?

Answer 7

The sequence you provided—1, 5, 25, 125—forms a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed, non-zero number, known as the common ratio.

In this case, the common ratio is 5, since:

• \(1 \times 5 = 5\)
• \(5 \times 5 = 25\)
• \(25 \times 5 = 125\)

Thus, each term in the sequence is multiplied by 5 to get the next term. Therefore, this is a geometric sequence with a common ratio of 5.

Answer 8

What kind of sequence is this?

3, –5, –11, –15, ...

Answer 9

The sequence you provided—3, -5, -11, -15—is an arithmetic sequence. In an arithmetic sequence, each term after the first is obtained by adding a constant difference to the previous term.

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

• Second term: \(-5 - 3 = -8\)
• Third term: \(-11 - (-5) = -6\)
• Fourth term: \(-15 - (-11) = -4\)

Upon checking, this sequence exhibits a varying difference in consecutive terms.

However, if we analyze the pattern closely:

• The difference from 3 to -5 is -8.
• The difference from -5 to -11 is -6.
• The difference from -11 to -15 is -4.

The differences between consecutive terms are changing, indicating it is not an arithmetic sequence.

This sequence does not fit standard patterns of arithmetic or geometric. In this case, it appears to be a non-standard sequence.

If you notice any additional pattern or rule governing the sequence, please let me know, and I can provide further analysis!

Answer 10

What kind of sequence is this?

56, 62, 66, 68, ...

Answer 11

The sequence you provided—56, 62, 66, 68—is an arithmetic sequence. In an arithmetic sequence, each term after the first is obtained by adding a constant difference to the previous term.

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

• From 56 to 62: \(62 - 56 = 6\)
• From 62 to 66: \(66 - 62 = 4\)
• From 66 to 68: \(68 - 66 = 2\)

We can see that the difference is not constant, as it decreases with each step:

• The first difference is 6.
• The second difference is 4.
• The third difference is 2.

Since the differences are changing, this sequence does not fit the definition of an arithmetic sequence.

However, it may indicate a different kind of pattern, potentially a quadratic or polynomial sequence, but further analysis would be necessary to identify it accurately.

If you're looking for explicit classification based solely on the values provided, we can conclude that it is neither strictly arithmetic nor geometric due to the varying differences. If you have more terms or context about the rule governing the sequence, please provide that for a more accurate classification!