In quantum sensing, a researcher models the sensitivity of a sensor as \( S(n) = \frac{100}{n^2 + 1} \) when measuring at discrete time intervals \(n = 1, 2, 3, \ldots\) seconds. At what integer time \(n\) is the sensitivity first less than 5 units?

["When is Sensor Sensitivity First Below 5 Units? A Quantum Sensing Model", "In quantum sensing, the precision of a sensor is often measured by its sensitivity, defined mathematically as ( S(n) = \frac{100}{n^2 + 1} ), where ( n ) represents time in seconds — discrete intervals starting from ( n = 1 ). Understanding when this sensitivity drops below a usable threshold — such as 5 units — is essential for optimizing measurement timing in quantum experiments.", "This article models the sensitivity function and determines the smallest integer ( n ) for which ( S(n) < 5 ).", "### The Sensitivity Function", "We are given:", "[\nS(n) = \frac{100}{n^2 + 1}\n]", "We seek the smallest integer ( n \geq 1 ) such that:", "[\n\frac{100}{n^2 + 1} < 5\n]", "### Solving the Inequality", "Start by solving the mathematical inequality:", "[\n\frac{100}{n^2 + 1} < 5\n]", "Multiply both sides by ( n^2 + 1 ) (which is always positive for real ( n )):", "[\n100 < 5(n^2 + 1)\n]", "Divide both sides by 5:", "[\n20 < n^2 + 1\n]", "Subtract 1:", "[\n19 < n^2\n]", "Take square roots (noting ( n > 0 )):", "[\nn > \sqrt{19}\n]", "Since ( \sqrt{16} = 4 ) and ( \sqrt{25} = 5 ), and ( 19 ) lies between them, approximate:", "[\n\sqrt{19} \approx 4.36\n]", "Thus, the smallest integer ( n ) satisfying ( n > \sqrt{19} ) is:", "[\nn = 5\n]", "### Verification", "Check ( S(4) ) and ( S(5) ) to confirm:", "- For ( n = 4 ):\n ( S(4) = \frac{100}{4^2 + 1} = \frac{100}{17} \approx 5.88 > 5 )", "- For ( n = 5 ):\n ( S(5) = \frac{100}{5^2 + 1} = \frac{100}{26} \approx 3.85 < 5 )", "Hence, ( n = 5 ) is indeed the first integer at which sensitivity drops below 5.", "### Conclusion", "In quantum sensing applications using this model, the first integer time at which sensor sensitivity falls below 5 units is ( n = 5 ) seconds. This result helps researchers time their measurements precisely, balancing precision and signal strength in quantum experiments.", "For further insight, exploring how sensitivity degrades with time enables better calibration and longer, more reliable quantum sensing protocols.", "---", "Keywords: quantum sensing, sensor sensitivity model, ( S(n) = \frac{100}{n^2 + 1} ), find n where ( S(n) < 5 ), discrete time intervals, quantum measurement precision, smallest integer threshold crossing."]