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While sorting some buttons, Helen put 448 buttons in the first box, 445 buttons in the second box, 439 buttons in the third box, and 430 buttons in the fourth box. What kind of sequence is this?

3 answers

To determine the kind of sequence formed by the number of buttons in each box, we can analyze the numbers: 448, 445, 439, and 430.

We can examine the differences between consecutive terms:

  • From the first box to the second box: \( 448 - 445 = 3 \)
  • From the second box to the third box: \( 445 - 439 = 6 \)
  • From the third box to the fourth box: \( 439 - 430 = 9 \)

Now, looking at the differences: 3, 6, and 9, we see that these differences themselves form an arithmetic sequence, increasing by 3 each time.

Since the differences between the terms are not constant but follow a consistent pattern (specifically, they increase by 3), we can deduce that the original sequence of button counts forms a quadratic sequence (also known as a second-degree polynomial sequence).

In conclusion, the sequence of buttons in the boxes is a quadratic sequence.

is is artimitic or geometiric?

Based on the analysis of the button counts:

  • The numbers of buttons in each box are: 448, 445, 439, and 430.

  • The differences between consecutive terms are:

    • \( 445 - 448 = -3 \)
    • \( 439 - 445 = -6 \)
    • \( 430 - 439 = -9 \)

These differences are not constant, but let's analyze them as we did before:

  1. The first difference: -3
  2. The second difference: -6
  3. The third difference: -9

The differences between terms are changing by a consistent subtraction (the increase in absolute value is 3 each time). This indicates the sequence does not form an arithmetic sequence (which would have constant differences) or a geometric sequence (which would have a constant ratio).

Therefore, the sequence is neither arithmetic nor geometric. Instead, it shows characteristics of a quadratic sequence due to the nature of the differences between terms changing consistently by a fixed amount.

To summarize:

  • Not Arithmetic: The differences between terms are not constant.
  • Not Geometric: The ratios between terms are not constant.

So the sequence does not fit into either category.