https://file.aiquickdraw.com/mathbot/file/1722997178005k191sdc6.jpg

Let's analyze the sequences provided in the image and find their general terms.

1. **Sequence 1**: \( 12, 14, 40, 82, 164 \)

   Let's find the differences between the terms:
   - \( 14 - 12 = 2 \)
   - \( 40 - 14 = 26 \)
   - \( 82 - 40 = 42 \)
   - \( 164 - 82 = 82 \)

   Differences:
   - \( 2, 26, 42, 82 \)

   It appears that the differences do not form a consistent pattern. Let's consider another approach, such as looking at the ratios or second differences.

   Ratios:
   - \( \frac{14}{12} = \frac{7}{6} \)
   - \( \frac{40}{14} = \frac{20}{7} \)
   - \( \frac{82}{40} = \frac{41}{20} \)
   - \( \frac{164}{82} = 2 \)

   Given the irregular pattern, it might be easier to identify if we look for a polynomial or exponential pattern. However, without a clear pattern emerging, let's leave this sequence for now and proceed with the others.

2. **Sequence 2**: \( 6, 10, 42, 208, 18.4 \)

   Differences:
   - \( 10 - 6 = 4 \)
   - \( 42 - 10 = 32 \)
   - \( 208 - 42 = 166 \)
   - \( 18.4 - 208 = -189.6 \)

   Similar to the first sequence, the differences are not consistent.

   Ratios:
   - \( \frac{10}{6} = \frac{5}{3} \)
   - \( \frac{42}{10} = 4.2 \)
   - \( \frac{208}{42} \approx 4.952 \)
   - \( \frac{18.4}{208} \approx 0.088 \)

   The ratios also do not form a clear pattern. Let's consider looking at polynomial or other patterns later.

3. **Sequence 3**: \( 9, 15, 23, 33, 45 \)

   Differences:
   - \( 15 - 9 = 6 \)
   - \( 23 - 15 = 8 \)
   - \( 33 - 23 = 10 \)
   - \( 45 - 33 = 12 \)

   The differences are \( 6, 8, 10, 12 \), which increase by 2 each time. This indicates a quadratic sequence.

   Let's express this as a quadratic sequence:
   \[
   a_n = an^2 + bn + c
   \]

   We can set up equations using the first three terms:
   \[
   \begin{cases}
   a(1)^2 + b(1) + c = 9 \\
   a(2)^2 + b(2) + c = 15 \\
   a(3)^2 + b(3) + c = 23 \\
   \end{cases}
   \]

   Simplifying these, we get:
   \[
   \begin{cases}
   a + b + c = 9 \\
   4a + 2b + c = 15 \\
   9a + 3b + c = 23 \\
   \end{cases}
   \]

   Subtract the first equation from the second:
   \[
   3a + b = 6 \quad \text{(Equation 1)}
   \]

   Subtract the second equation from the third:
   \[
   5a + b = 8 \quad \text{(Equation 2)}
   \]

   Subtract Equation 1 from Equation 2:
   \[
   2a = 2 \implies a = 1
   \]

   Substitute \( a = 1 \) back into Equation 1:
   \[
   3(1) + b = 6 \implies b = 3
   \]

   Substitute \( a = 1 \) and \( b = 3 \) back into the first equation:
   \[
   1 + 3 + c = 9 \implies c = 5
   \]

   Therefore, the general term for Sequence 3 is:
   \[
   a_n = n^2 + 3n + 5
   \]

Would you like to go through the details for the other sequences, or do you have any questions?

Here are some related questions you might find useful:
1. How do you find the general term of an arithmetic sequence?
2. What is the difference between arithmetic and geometric sequences?
3. How can you identify a quadratic sequence?
4. What methods can be used to find the general term of a sequence?
5. How do you solve a system of equations to find the coefficients of a polynomial?
6. What are common sequences encountered in mathematics?
7. How can you verify the general term of a sequence?
8. How do polynomial sequences differ from exponential sequences?

**Tip:** Always check for common patterns such as constant differences (arithmetic), constant ratios (geometric), or polynomial forms when identifying the general term of a sequence.

Learn how to find the general term of sequences with a step-by-step solution. This problem covers key concepts in sequences, including polynomial patterns and quadratic sequences. Suitable for high school students studying mathematics.

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