Question

6.
Use inductive reasoning to describe the pattern. Then find the next two numbers in the pattern.

–3, 9, –27, 81, . . .

Answer 1

To identify the pattern in the sequence \(-3, 9, -27, 81, \ldots\), let's analyze the relationship between the numbers:

1. Start with \(-3\).
2. The next number is \(9\), which can be obtained from \(-3 \times -3 = 9\).
3. The next number is \(-27\), which can be calculated from \(9 \times -3 = -27\).
4. The next number is \(81\), which comes from \(-27 \times -3 = 81\).

From this, we can see that the pattern involves multiplying by \(-3\) alternately. Therefore, the sequence can be described as follows:

- Start with \(-3\)
- Multiply by \(-3\) to get \(9\)
- Multiply by \(-3\) to get \(-27\)
- Multiply by \(-3\) to get \(81\)

Continuing this pattern:
- Multiply \(81\) by \(-3\) to get \(-243\).
- Multiply \(-243\) by \(-3\) to get \(729\).

Thus, the next two numbers in the pattern are \(-243\) and \(729\).

In conclusion, the next two numbers are:
\(-243, 729\).