a sequence of numbers follows the rule: multiply the previous number by -2 and add 3, the fourth term in the sequence is -7

Let's determine the first three terms of the sequence using the rule: multiply the previous number by \(-2\) and add \(3\).

Let the first term be \(a_1\), the second term be \(a_2\), and so on. We know:

\[
a_4 = -7
\]

To find the terms that came before \(a_4 = -7\), we work backwards:

1. **Find \(a_3\)**:  
   \[
   a_4 = -2 \times a_3 + 3
   \]
   Substitute \(a_4 = -7\):
   \[
   -7 = -2 \times a_3 + 3
   \]
   Solve for \(a_3\):
   \[
   -10 = -2 \times a_3 \implies a_3 = 5
   \]

2. **Find \(a_2\)**:  
   \[
   a_3 = -2 \times a_2 + 3
   \]
   Substitute \(a_3 = 5\):
   \[
   5 = -2 \times a_2 + 3
   \]
   Solve for \(a_2\):
   \[
   2 = -2 \times a_2 \implies a_2 = -1
   \]

3. **Find \(a_1\)**:  
   \[
   a_2 = -2 \times a_1 + 3
   \]
   Substitute \(a_2 = -1\):
   \[
   -1 = -2 \times a_1 + 3
   \]
   Solve for \(a_1\):
   \[
   -4 = -2 \times a_1 \implies a_1 = 2
   \]

So, the first three terms in the sequence are:

1. \(a_1 = 2\)
2. \(a_2 = -1\)
3. \(a_3 = 5\)

The three terms that come before \(-7\) are **2, -1,** and **5**.

Would you like more details or have any questions? Here are some related questions:

1. What is the fifth term in this sequence?
2. How do you find the general formula for the \(n\)th term in such sequences?
3. How does changing the multiplier from \(-2\) to another number affect the sequence?
4. What if the rule was "divide by 2 and subtract 3" instead?
5. Can you find the sum of the first 10 terms in this sequence?

**Tip:** Always verify your final answers by plugging them back into the original rule to ensure consistency.

Learn how to solve problems involving arithmetic sequences where each term is derived from the previous term by multiplying by -2 and adding 3. This problem specifically addresses finding specific terms and developing a general formula. Suitable for students in Grades 7-9.

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