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t = \frac{4 \pm 2}{6}
t = \frac{6}{6} = 1 \quad \text{and} \quad t = \frac{2}{6} = \frac{1}{3}
Both values satisfy the equation. Thus, the values of \(t\) are:
\boxed{1 \quad \text{and} \quad \frac{1}{3}}
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}
We attempt polynomial division or rewrite the numerator:
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}