To determine if the given sequence $1, 2, 4$ can be quadratic, we assume that the terms of the sequence follow the general quadratic form:

$a_n = An^2 + Bn + C$

where $A$, $B$, and $C$ are constants. Using the first three terms, we can set up a system of equations based on the given values:

1. For $n = 1$, $a_1 = 1$: $A(1)^2 + B(1) + C = 1 \quad \Rightarrow \quad A + B + C = 1$

2. For $n = 2$, $a_2 = 2$: $A(2)^2 + B(2) + C = 2 \quad \Rightarrow \quad 4A + 2B + C = 2$

3. For $n = 3$, $a_3 = 4$: $A(3)^2 + B(3) + C = 4 \quad \Rightarrow \quad 9A + 3B + C = 4$

We now solve this system of equations.

Step 1: Subtract equations to eliminate $C$

From $A + B + C = 1$: $C = 1 - A - B$

Substitute $C$ into the other two equations:

1. $4A + 2B + C = 2$: $4A + 2B + (1 - A - B) = 2 \quad \Rightarrow \quad 3A + B = 1$

2. $9A + 3B + C = 4$: $9A + 3B + (1 - A - B) = 4 \quad \Rightarrow \quad 8A + 2B = 3$

Step 2: Solve for $A$ and $B$

From $3A + B = 1$, solve for $B$: $B = 1 - 3A$

Substitute $B = 1 - 3A$ into $8A + 2B = 3$: $8A + 2(1 - 3A) = 3 \quad \Rightarrow \quad 8A + 2 - 6A = 3 \quad \Rightarrow \quad 2A = 1 \quad \Rightarrow \quad A = \frac{1}{2}$

Substitute $A = \frac{1}{2}$ into $B = 1 - 3A$: $B = 1 - 3\left(\frac{1}{2}\right) = 1 - \frac{3}{2} = -\frac{1}{2}$

Step 3: Solve for $C$

Substitute $A = \frac{1}{2}$ and $B = -\frac{1}{2}$ into $C = 1 - A - B$: $C = 1 - \frac{1}{2} - \left(-\frac{1}{2}\right) = 1 - \frac{1}{2} + \frac{1}{2} = 1$

Step 4: Find the 4th term

The general quadratic equation is: $a_n = \frac{1}{2}n^2 - \frac{1}{2}n + 1$

For $n = 4$: $a_4 = \frac{1}{2}(4)^2 - \frac{1}{2}(4) + 1 = \frac{1}{2}(16) - 2 + 1 = 8 - 2 + 1 = 7$

Conclusion

Yes, the sequence can be quadratic. The 4th term is:

$\boxed{7}$

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Would you like more details or clarifications?
Here are 5 related questions to expand on this:

1. How can you determine whether a sequence is linear, quadratic, or higher-order?
2. What are the conditions for a quadratic sequence to exist?
3. Can we extend this process to find the $n$-th term for this sequence?
4. How does the method change if the first three terms do not suggest a quadratic sequence?
5. Can a sequence have more than one possible quadratic representation?

Tip: When working with sequences, analyzing the differences between consecutive terms can quickly suggest whether the sequence might be quadratic or higher-order!