Questions: Learn with an example Which word best describes this formula for the sequence (an) ? (an=5 an-2+3 an-3+10 n^2-15)

Learn with an example

Which word best describes this formula for the sequence (an) ?
(an=5 an-2+3 an-3+10 n^2-15)
Transcript text: Learn with an example Which word best describes this formula for the sequence $a_{n}$ ? \[ a_{n}=5 a_{n-2}+3 a_{n-3}+10 n^{2}-15 \]
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Solution

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Solution Steps

To determine the type of sequence described by the given formula, we need to analyze the recurrence relation. The formula involves terms that depend on previous terms in the sequence, as well as a polynomial term in \( n \). This suggests that the sequence is defined recursively with additional polynomial growth.

Step 1: Identify the Type of Sequence

The given formula for the sequence \( a_n \) is: \[ a_n = 5a_{n-2} + 3a_{n-3} + 10n^2 - 15 \] This formula involves terms that depend on previous terms in the sequence (\( a_{n-2} \) and \( a_{n-3} \)), as well as a polynomial term in \( n \) (\( 10n^2 - 15 \)).

Step 2: Classify the Sequence

Since the sequence is defined using previous terms and includes a polynomial term, it is best described as a recurrence relation with a polynomial term.

Final Answer

\[ \boxed{\text{Recurrence relation with polynomial term}} \]

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