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