Math Problem Statement
What is the equation in the sequence 0, 37, 106, 235, 442, 745
Solution
To find the equation of the sequence , we need to determine a pattern or rule that relates each term to its position in the sequence.
Let's denote the -th term of the sequence as .
The first step is to check the differences between consecutive terms:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Sequences
Polynomial equations
Pattern recognition
Formulas
Cubic equation form
Theorems
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Suitable Grade Level
Grades 9-12
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