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The sequence provided is quadratic, meaning its \(n\)th term can be expressed in the form:

\[
n^2 + cn
\]

where \(c\) is a constant to be determined. 

From the table:

| Position \(n\) | 1  | 2  | 3  | 4  | 5  |
|----------------|----|----|----|----|----|
| Term           | 4  | 10 | 18 | 28 | 40 |

The general formula starts with \(n^2\), so we first calculate \(n^2\) for each value of \(n\):

- For \(n = 1\): \(n^2 = 1\)
- For \(n = 2\): \(n^2 = 4\)
- For \(n = 3\): \(n^2 = 9\)
- For \(n = 4\): \(n^2 = 16\)
- For \(n = 5\): \(n^2 = 25\)

Next, subtract \(n^2\) from each corresponding term to find the remainder:

- For \(n = 1\): \(4 - 1 = 3\)
- For \(n = 2\): \(10 - 4 = 6\)
- For \(n = 3\): \(18 - 9 = 9\)
- For \(n = 4\): \(28 - 16 = 12\)
- For \(n = 5\): \(40 - 25 = 15\)

This shows that the difference between the term and \(n^2\) is increasing by 3 each time, which suggests a linear relationship. The additional term is \(3n\).

Thus, the \(n\)th term rule for this sequence is:

\[
n^2 + 3n
\]

Would you like any further clarification?

Here are 5 related questions:
1. What is the next term in this quadratic sequence?
2. How do you find the difference between terms in a quadratic sequence?
3. What role does the constant \(c\) play in the general formula of a quadratic sequence?
4. Can you derive a general method for finding the quadratic sequence rule from any given set of terms?
5. How would this method apply to sequences that are not quadratic but cubic?

**Tip**: When analyzing sequences, always check if differences between terms are constant or if they follow a pattern that hints at a higher-degree polynomial relationship.

Complete the nth term rule for the quadratic sequence where the positions and corresponding terms are given.

Learn how to derive the nth term rule for a given quadratic sequence with positions 1 to 5 and terms like 4, 10, 18, 28, and 40. This problem involves understanding quadratic sequences and using algebraic techniques to determine the missing constant in the formula.

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