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To determine the $n$th term of the sequence $4, 10, 18, 28$, we can use the formula $n^2 + 3n$.

To derive the $n$th term of a sequence, we first need to identify the underlying pattern or rule that generates the terms. For the sequence given, we analyze the differences between consecutive terms:

• The first differences are:
  • $10 - 4 = 6$
  • $18 - 10 = 8$
  • $28 - 18 = 10$

These first differences ($6, 8, 10$) are not constant, so we calculate the second differences:

• The second differences are:
  • $8 - 6 = 2$
  • $10 - 8 = 2$

Since the second differences are constant, we can conclude that this is a quadratic sequence.

A quadratic sequence can be represented in the general form $an^2 + bn + c$. To find the coefficients $a$, $b$, and $c$, we will use the first few terms of the sequence.

We establish a system of equations based on the terms of the sequence:

1. For $n = 1$: $a(1)^2 + b(1) + c = 4$ which simplifies to $a + b + c = 4$

2. For $n = 2$: $a(2)^2 + b(2) + c = 10$ simplifying to $4a + 2b + c = 10$

3. For $n = 3$: $a(3)^2 + b(3) + c = 18$ simplifying to $9a + 3b + c = 18$

This gives us the following system of equations:

1. $a + b + c = 4$
2. $4a + 2b + c = 10$
3. $9a + 3b + c = 18$

Next, we will solve these equations.

First, we subtract the first equation from the second:

$(4a + 2b + c) - (a + b + c) = 10 - 4$ This simplifies to: $3a + b = 6 \quad \text{(Equation 4)}$

Next, we subtract the second equation from the third:

$(9a + 3b + c) - (4a + 2b + c) = 18 - 10$ This simplifies to: $5a + b = 8 \quad \text{(Equation 5)}$

Now, we can eliminate $b$ by subtracting Equation 4 from Equation 5:

$(5a + b) - (3a + b) = 8 - 6$ This simplifies to: $2a = 2 \implies a = 1$

Next, we substitute $a = 1$ back into Equation 4 to find $b$:

$3(1) + b = 6$ This simplifies to: $3 + b = 6 \implies b = 3$

Finally, we substitute both $a = 1$ and $b = 3$ back into the first equation to find $c$:

$1 + 3 + c = 4$ This simplifies to: $4 + c = 4 \implies c = 0$

Thus, the formula for the $n$th term of the sequence is given by:

$n^2 + 3n$