To identify the pattern in the sequence \(0, 1, 10, 57, ?\), we need to find the relation between the terms. Let's examine the differences between consecutive terms to see if there's a pattern:

1. \(1 - 0 = 1\)
2. \(10 - 1 = 9\)
3. \(57 - 10 = 47\)

Next, let's see if there is a pattern in these differences:
1. \(1\)
2. \(9\)
3. \(47\)

Since the differences do not seem to follow a simple pattern, let's consider other possibilities. One way to approach this problem is to see if each term can be represented as a polynomial function of \(n\), where \(n\) is the position of the term in the sequence starting from \(0\).

Let's assume \(a_n = An^3 + Bn^2 + Cn + D\). We need to find \(A\), \(B\), \(C\), and \(D\) such that:
- \(a_0 = 0\)
- \(a_1 = 1\)
- \(a_2 = 10\)
- \(a_3 = 57\)

Substituting the values, we get:
1. \(0 = A(0)^3 + B(0)^2 + C(0) + D \Rightarrow D = 0\)
2. \(1 = A(1)^3 + B(1)^2 + C(1) + D \Rightarrow A + B + C + 0 = 1\)
3. \(10 = A(2)^3 + B(2)^2 + C(2) + D \Rightarrow 8A + 4B + 2C + 0 = 10\)
4. \(57 = A(3)^3 + B(3)^2 + C(3) + D \Rightarrow 27A + 9B + 3C + 0 = 57\)

This gives us the following system of equations:
1. \(A + B + C = 1\)
2. \(8A + 4B + 2C = 10\)
3. \(27A + 9B + 3C = 57\)

We can solve this system step-by-step.

Let's first simplify the second equation by dividing by 2:
\[4A + 2B + C = 5\]

Next, we can subtract the first equation from the simplified second equation to eliminate \(C\):
\[ (4A + 2B + C) - (A + B + C) = 5 - 1\]
\[ 3A + B = 4\]

Now, let's simplify the third equation by dividing by 3:
\[9A + 3B + C = 19\]

Next, subtract the first equation from the simplified third equation to eliminate \(C\):
\[ (9A + 3B + C) - (A + B + C) = 19 - 1\]
\[ 8A + 2B = 18\]
\[ 4A + B = 9\]

Now we have a new system:
1. \(3A + B = 4\)
2. \(4A + B = 9\)

Subtract the first equation from the second to eliminate \(B\):
\[ (4A + B) - (3A + B) = 9 - 4\]
\[ A = 5\]

Now substitute \(A = 5\) into the first equation:
\[ 3(5) + B = 4\]
\[ 15 + B = 4\]
\[ B = -11\]

Finally, substitute \(A = 5\) and \(B = -11\) back into the first system:
\[ 5 - 11 + C = 1\]
\[ -6 + C = 1\]
\[ C = 7\]

So, the polynomial that represents the sequence is:
\[ a_n = 5n^3 - 11n^2 + 7n \]

Now, we can find the next term \(a_4\):
\[ a_4 = 5(4)^3 - 11(4)^2 + 7(4) \]
\[ a_4 = 5(64) - 11(16) + 28 \]
\[ a_4 = 320 - 176 + 28 \]
\[ a_4 = 172 \]

Therefore, the next term in the sequence is \(172\).

Would you like any more details or have any questions?

Here are 8 related questions to expand on this information:
1. Can you explain how to derive a polynomial function for a given sequence?
2. What are some other methods to identify patterns in sequences?
3. How can you verify if a polynomial function correctly represents a sequence?
4. What are the common types of sequences in mathematics?
5. How can differences between terms in a sequence help in identifying patterns?
6. What are some applications of polynomial functions in real life?
7. How can you determine the degree of the polynomial needed to represent a sequence?
8. What are the limitations of using polynomial functions to represent sequences?

**Tip:** When solving systems of equations, consistently check your calculations at each step to avoid errors.

Explore how to identify the next term in the sequence 0, 1, 10, 57, ?. Learn about polynomial functions and sequence analysis with a detailed step-by-step solution. Suitable for students in Grades 10-12.

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