A hospital treats 4 patients in the first hour, 8 patients in the second hour, 12 patients in the third hour and so on. [patients treated at nth hour is 4 more than the patients treated in \((n-1)\) th hour]. How many hours will it require to treat a total of 84 patients?

Explanation: The number of patients treated each hour is an arithmetic progression: \(4, 8, 12, 16, 20, 24, \dots\). The sum of patients treated after \(n\) hours is \(S_n = \frac{n}{2}[2a + (n-1)d]\). Here \(a=4\), \(d=4\), and \(S_n=84\). \(84 = \frac{n}{2}[2(4) + (n-1)4] \Rightarrow 168 = n[8 + 4n - 4] \Rightarrow 168 = n[4n+4] \Rightarrow 168 = 4n^2 + 4n \Rightarrow 4n^2 + 4n - 168 = 0 \Rightarrow n^2 + n - 42 = 0 \Rightarrow (n+7)(n-6) = 0\). Since \(n\) must be positive, \(n=6\). Alternatively: \(4 + 8 + 12 + 16 + 20 + 24 = 84\)। সুতরাং, 84 জন patient treat করতে লাগবে 6 hours।