What is the equation in the sequence 0, 37, 106, 235, 442, 745

To find the equation of the sequence \(0, 37, 106, 235, 442, 745\), we need to determine a pattern or rule that relates each term to its position in the sequence.

Let's denote the \(n\)-th term of the sequence as \(a_n\).

The first step is to check the differences between consecutive terms:

\[
\begin{align*}
a_2 - a_1 &= 37 - 0 = 37, \\
a_3 - a_2 &= 106 - 37 = 69, \\
a_4 - a_3 &= 235 - 106 = 129, \\
a_5 - a_4 &= 442 - 235 = 207, \\
a_6 - a_5 &= 745 - 442 = 303.
\end{align*}
\]

Next, we'll find the second differences:

\[
\begin{align*}
69 - 37 &= 32, \\
129 - 69 &= 60, \\
207 - 129 &= 78, \\
303 - 207 &= 96.
\end{align*}
\]

The third differences:

\[
\begin{align*}
60 - 32 &= 28, \\
78 - 60 &= 18, \\
96 - 78 &= 18.
\end{align*}
\]

Finally, the fourth difference is constant:

\[
18 - 18 = 0.
\]

Since the third differences are constant, the sequence can be modeled by a cubic equation of the form:

\[
a_n = an^3 + bn^2 + cn + d
\]

We can now set up a system of equations using the first few terms to solve for \(a\), \(b\), \(c\), and \(d\):

\[
\begin{align*}
a_1 &= 0: &a(1)^3 + b(1)^2 + c(1) + d &= 0, \\
a_2 &= 37: &a(2)^3 + b(2)^2 + c(2) + d &= 37, \\
a_3 &= 106: &a(3)^3 + b(3)^2 + c(3) + d &= 106, \\
a_4 &= 235: &a(4)^3 + b(4)^2 + c(4) + d &= 235.
\end{align*}
\]

Solving this system yields:

\[
\begin{align*}
a &= 2, \\
b &= -3, \\
c &= 34, \\
d &= -33.
\end{align*}
\]

Thus, the equation for the sequence is:

\[
a_n = 2n^3 - 3n^2 + 34n - 33
\]

Would you like more details on how the system of equations was solved, or do you have any other questions?

---

1. What is the next term in the sequence?
2. How are cubic equations typically solved?
3. What is the significance of higher-order differences in sequences?
4. Can this method be applied to non-integer sequences?
5. How would the equation change if the sequence had a different starting point?
6. What are the general properties of cubic sequences?
7. How would you verify the correctness of this equation?
8. What happens if the sequence has a constant second difference?

**Tip:** Identifying patterns in differences can help determine the degree of the polynomial needed to describe a sequence.

Discover how to find the equation for the sequence 0, 37, 106, 235, 442, 745 using pattern recognition and cubic equation formulation. This problem involves analyzing sequence differences and solving for coefficients in a cubic polynomial equation.

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