To find the equation of the sequence $2, 21, 66, 149$, we need to identify a pattern or use a method like finite differences or fitting a polynomial.

Method: Finite Differences

1. First Differences:

   • $21 - 2 = 19$
   • $66 - 21 = 45$
   • $149 - 66 = 83$ First differences: $19, 45, 83$

2. Second Differences:

   • $45 - 19 = 26$
   • $83 - 45 = 38$ Second differences: $26, 38$

3. Third Differences:

   • $38 - 26 = 12$ Third difference: $12$

Since the third differences are constant, the sequence can be represented by a cubic polynomial of the form: $a_n = an^3 + bn^2 + cn + d$ Now, substitute the values of $n$ and $a_n$ (where $n$ is the position in the sequence, and $a_n$ is the sequence value):

• $a_1 = 2$ for $n = 1$
• $a_2 = 21$ for $n = 2$
• $a_3 = 66$ for $n = 3$
• $a_4 = 149$ for $n = 4$

We'll form the system of equations to solve for $a$, $b$, $c$, and $d$.

1. For $n = 1$: $a(1)^3 + b(1)^2 + c(1) + d = 2 \implies a + b + c + d = 2$
2. For $n = 2$: $a(2)^3 + b(2)^2 + c(2) + d = 21 \implies 8a + 4b + 2c + d = 21$
3. For $n = 3$: $a(3)^3 + b(3)^2 + c(3) + d = 66 \implies 27a + 9b + 3c + d = 66$
4. For $n = 4$: $a(4)^3 + b(4)^2 + c(4) + d = 149 \implies 64a + 16b + 4c + d = 149$

Let's solve this system to find the values of $a$, $b$, $c$, and $d$.The solution to the system of equations gives us the values:

• $a = 2$
• $b = 1$
• $c = 2$
• $d = -3$

So, the equation for the sequence is:

$a_n = 2n^3 + n^2 + 2n - 3$

This equation should generate the sequence $2, 21, 66, 149$ when $n = 1, 2, 3, 4$.

Would you like more details, or do you have any questions?

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Here are 8 related questions:

1. How can you verify that the sequence fits the equation $2n^3 + n^2 + 2n - 3$?
2. What is the next term in the sequence using the equation?
3. How would the sequence change if $d$ were positive?
4. Can we derive a general formula for any cubic sequence?
5. How do finite differences help in finding the polynomial equation?
6. What if the third difference wasn't constant? What would that indicate?
7. How can this approach be extended to higher-degree polynomials?
8. What are the limitations of using polynomial fitting for sequences?

Tip: When working with sequences, identifying the degree of the polynomial is crucial to finding the correct pattern or rule.