We attempt polynomial division or rewrite the numerator:
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- Question:** A sequence of neural response intensities in a human-machine interface is defined by \(a_n = \frac{n^2 + 1}{n + 1}\). Simplify \(a_n\) and find the value when \(n = 4\).
- We simplify the expression:
- a_n = \frac{n^2 + 1}{n + 1}
- Note that \(n^2 + 1\) does not factor nicely over integers, so perform direct substitution for \(n = 4\):
- a_4 = \frac{4^2 + 1}{4 + 1} = \frac{16 + 1}{5} = \frac{17}{5}
- Thus, the simplified form evaluated at \(n = 4\) is:
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