Question

Which best describes the rule for this pattern?

1, –4, 3, –2, . . .

Answer 1

To understand the pattern of the sequence \(1, -4, 3, -2, \ldots\), we can look at the differences between consecutive terms:

1. The first term is \(1\).
2. The second term is \(-4\).
- Difference: \(-4 - 1 = -5\)
3. The third term is \(3\).
- Difference: \(3 - (-4) = 3 + 4 = 7\)
4. The fourth term is \(-2\).
- Difference: \(-2 - 3 = -5\)

Now let's summarize the process:

- From the first term to the second, we subtract \(5\).
- From the second term to the third, we add \(7\).
- From the third term to the fourth, we subtract \(5\).

We can see that the pattern alternates between subtracting \(5\) and adding \(7\).

Using this alternating pattern, we can predict the next term:

- From \(-2\), we add \(7\):

\(-2 + 7 = 5\)

Thus, the next term would be \(5\), and we can also suggest that the next term after that would subtract \(5\) from \(5\):

So, the rule can be described as follows:

1. Start with \(1\).
2. Alternate between subtracting \(5\) and adding \(7\).

The pattern suggests that the next few numbers would continue to alternate, producing a sequence of:

1, -4, 3, -2, 5, 0, 7, etc.

In summary, the rule for the pattern is: **Subtract \(5\) from the previous term, then add \(7\) to the previous term alternately.**