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:

   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.