The given sequence is: 
1, 7, 19, 43, 91.

### Step 1: Identify the pattern
This is a sequence that seems to grow more rapidly with each step. To check, let's find the **differences** between consecutive terms:

\[
7 - 1 = 6 \\
19 - 7 = 12 \\
43 - 19 = 24 \\
91 - 43 = 48
\]

### Step 2: Check the second differences
Now, let's find the second differences (differences between the first differences):

\[
12 - 6 = 6 \\
24 - 12 = 12 \\
48 - 24 = 24
\]

The second differences are not constant, but let's check the third differences:

\[
12 - 6 = 6 \\
24 - 12 = 12
\]

The **third differences** are constant, which suggests this is a cubic sequence. Therefore, the explicit formula is likely of the form:

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

### Step 3: Solve for \(a\), \(b\), \(c\), and \(d\)
We will now use the given terms to set up a system of equations by plugging in values of \(n\) for each term in the sequence:

1. For \(n = 1\), \(a_1 = 1\):
\[
a(1^3) + b(1^2) + c(1) + d = 1 \quad \Rightarrow \quad a + b + c + d = 1
\]
2. For \(n = 2\), \(a_2 = 7\):
\[
a(2^3) + b(2^2) + c(2) + d = 7 \quad \Rightarrow \quad 8a + 4b + 2c + d = 7
\]
3. For \(n = 3\), \(a_3 = 19\):
\[
a(3^3) + b(3^2) + c(3) + d = 19 \quad \Rightarrow \quad 27a + 9b + 3c + d = 19
\]
4. For \(n = 4\), \(a_4 = 43\):
\[
a(4^3) + b(4^2) + c(4) + d = 43 \quad \Rightarrow \quad 64a + 16b + 4c + d = 43
\]

### Step 4: Solve the system of equations
Now, solve the system of four equations:

1. \(a + b + c + d = 1\)
2. \(8a + 4b + 2c + d = 7\)
3. \(27a + 9b + 3c + d = 19\)
4. \(64a + 16b + 4c + d = 43\)

By solving this system, we get:

\[
a = 2, \quad b = -3, \quad c = 0, \quad d = 2
\]

### Step 5: Write the explicit formula
Thus, the explicit formula for the sequence is:

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

This is the explicit formula for the given sequence.

Would you like further details or have any questions?

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Here are some related questions you might find helpful:
1. What is the next term in the sequence using the formula \(a_n = 2n^3 - 3n^2 + 2\)?
2. Can you find an explicit formula for a quadratic sequence?
3. How do you verify the explicit formula by plugging in values for \(n\)?
4. What are the differences between linear, quadratic, and cubic sequences?
5. How can you determine the degree of a sequence?

**Tip:** Recognizing the pattern of differences helps in identifying the degree of the polynomial formula for the sequence.

Learn how to derive the explicit formula for the cubic sequence 1, 7, 19, 43, 91 using the method of differences. This solution covers cubic sequences, polynomial functions, and step-by-step system of equations solving. Ideal for students in Grades 10-12.

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