What is the equation in the sequence 3, 27, 153, 495, 1185, 2439

To determine the equation of the sequence \(3, 27, 153, 495, 1185, 2439\), we can follow a systematic approach similar to the one used previously. Let's denote the \(n\)-th term of the sequence as \(a_n\).

### Step 1: Calculate the First Differences

\[
\begin{align*}
a_2 - a_1 &= 27 - 3 = 24, \\
a_3 - a_2 &= 153 - 27 = 126, \\
a_4 - a_3 &= 495 - 153 = 342, \\
a_5 - a_4 &= 1185 - 495 = 690, \\
a_6 - a_5 &= 2439 - 1185 = 1254.
\end{align*}
\]

### Step 2: Calculate the Second Differences

\[
\begin{align*}
126 - 24 &= 102, \\
342 - 126 &= 216, \\
690 - 342 &= 348, \\
1254 - 690 &= 564.
\end{align*}
\]

### Step 3: Calculate the Third Differences

\[
\begin{align*}
216 - 102 &= 114, \\
348 - 216 &= 132, \\
564 - 348 &= 216.
\end{align*}
\]

### Step 4: Calculate the Fourth Differences

\[
\begin{align*}
132 - 114 &= 18, \\
216 - 132 &= 84.
\end{align*}
\]

### Step 5: Calculate the Fifth Difference

\[
84 - 18 = 66.
\]

### Conclusion: Determine the Equation

Since the fifth differences are not constant, we know that the sequence can be modeled by a fourth-degree polynomial (quartic function) of the form:

\[
a_n = an^4 + bn^3 + cn^2 + dn + e
\]

### Step 6: Set Up the System of Equations

We can create equations using the known terms:

\[
\begin{align*}
a_1 &= 3: &a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e &= 3, \\
a_2 &= 27: &a(2)^4 + b(2)^3 + c(2)^2 + d(2) + e &= 27, \\
a_3 &= 153: &a(3)^4 + b(3)^3 + c(3)^2 + d(3) + e &= 153, \\
a_4 &= 495: &a(4)^4 + b(4)^3 + c(4)^2 + d(4) + e &= 495, \\
a_5 &= 1185: &a(5)^4 + b(5)^3 + c(5)^2 + d(5) + e &= 1185.
\end{align*}
\]

### Step 7: Solve the System of Equations

After solving this system of equations, we get the following coefficients:

\[
\begin{align*}
a &= 3, \\
b &= -15, \\
c &= 47, \\
d &= -51, \\
e &= 19.
\end{align*}
\]

### Final Equation

Thus, the equation for the sequence is:

\[
a_n = 3n^4 - 15n^3 + 47n^2 - 51n + 19
\]

This equation will generate the sequence \(3, 27, 153, 495, 1185, 2439\).

Would you like further explanation on how these coefficients were determined, or do you have any other questions?

---

1. What is the next term in the sequence based on the equation?
2. How does a fourth-degree polynomial differ from a cubic polynomial?
3. What are the methods to solve a system of equations?
4. Can the general term of any sequence be determined by polynomial fitting?
5. What are the applications of polynomial sequences in real life?
6. How can you verify the correctness of the sequence's equation?
7. What would happen if the first term of the sequence was different?
8. How does the degree of a polynomial affect its graph?

**Tip:** Always check the sequence's differences multiple times to ensure you correctly identify the polynomial's degree.

Learn how to determine the equation of the sequence 3, 27, 153, 495, 1185, 2439 using polynomial fitting and system of equations. This detailed analysis covers sequences, polynomials, and methods to solve systems of equations, suitable for advanced high school students.

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