**Problem Statement:**Use polynomial fitting to find the formula for the $n$th term of the sequence $(a_n)_{n \geq 0}$ which starts as follows: \[0, 0, 6, 18, 36, 60, \ldots\]---**Interactive Section:** ### Derive the FormulaUse the input box below to find the formula for the sequence:\[ a_n = \](Interactive input for the user to enter their solution.)---**Explanation:**This sequence begins with the terms 0, 0, 6, 18, 36, 60, … and the task is to determine a polynomial formula, $a_n$, that generates these terms for $n \geq 0$.---**Tips for Solution:**1. **Pattern Observation:** Check if the differences between terms suggest a particular degree of polynomial.2. **Polynomial Equation Setup:** Use known polynomial fitting techniques to derive coefficients.3. **Verification:** Substitute small values of $n$ to ensure they produce corresponding terms of the sequence.---This exercise encourages understanding polynomial fitting methods, often used in mathematical analysis to model sequences or series data.

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Description: **Problem Statement:**Use polynomial fitting to find the formula for the $n$th term of the sequence $(a_n)_{n \geq 0}$ which starts as follows: \[0, 0, 6, 18, 36, 60, \ldots\]---**Interactive Section:** ### Derive the FormulaUse the input box b

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se polynomial fitting to find the formula for the nth term of the sequence (an)n>0 which starts, 0,0, 6, 18, 36, 60, ...

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

**Problem Statement:**

Use polynomial fitting to find the formula for the $n$th term of the sequence $(a_n)_{n \geq 0}$ which starts as follows:

\[0, 0, 6, 18, 36, 60, \ldots\]

---

**Interactive Section:**
### Derive the Formula
Use the input box below to find the formula for the sequence:

\[ a_n = \]

(Interactive input for the user to enter their solution.)

---

**Explanation:**
This sequence begins with the terms 0, 0, 6, 18, 36, 60, … and the task is to determine a polynomial formula, $a_n$, that generates these terms for $n \geq 0$.

---

**Tips for Solution:**
1. **Pattern Observation:** Check if the differences between terms suggest a particular degree of polynomial.
2. **Polynomial Equation Setup:** Use known polynomial fitting techniques to derive coefficients.
3. **Verification:** Substitute small values of $n$ to ensure they produce corresponding terms of the sequence.

---

This exercise encourages understanding polynomial fitting methods, often used in mathematical analysis to model sequences or series data.

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