A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $\mathrm{m}$ is equal to:

<p>To determine the value of $\mathrm{m}$, we need to formulate the problem using some basic concepts of arithmetic progression and work. Let's first understand the nature of the problem:</p> <p>Initially, there are $\mathrm{m}$ computers, and it is estimated that with these $\mathrm{m}$ computers, the assignment can be completed in 17 days.</p> <p>However, due to the crash of 4 computers every day starting from the second day onward, the total time taken extends by 8 days, making it 25 days in total.</p> <p>To begin with, let's define the total work (W) in terms of the number of computers and days:</p> <p>The total work (W) is given by:</p> <p>The amount of work completed each day with $\mathrm{m}$ computers for 17 days:</p> <p>$W = 17m$</p> <p>When computers crash, the number of working computers each day forms an arithmetic sequence. On the first day, there are $\mathrm{m}$ computers. On the second day, there are $\mathrm{m} - 4$ computers, on the third day, there are $\mathrm{m} - 8$ computers, and so on. We need to sum this series until 25 days are completed.</p> <p>This can be formulated as:</p> <p>Total work done over 25 days with decrement in the number of computers:</p> <p>$W = m + (m - 4) + (m - 8) + \ldots + \left[m - 4 \times (n - 1)\right]$</p> <p>where $n$ is the number of days. Here, $n = 25$.</p> <p>Notice that we form an arithmetic series where the first term (a) is $\mathrm{m}$ and the common difference (d) is -4. The sum of the first n terms of an arithmetic series is:</p> <p>$S_n = \frac{n}{2} \left[ 2a + (n - 1)d \right]$</p> <p>Plugging in the values:</p> <p>$S_{25} = \frac{25}{2} \left[ 2m + (25 - 1)(-4) \right]$</p> <p>$S_{25} = \frac{25}{2} \left[ 2m - 96 \right]$</p> <p>$S_{25} = \frac{25}{2} \left[ 2m - 96 \right] = 25(m - 48)$</p> <p>This work should be equivalent to the work calculated earlier, so:</p> <p>$17m = 25(m - 48)$</p> <p>Solving for $\mathrm{m}$:</p> <p>$17m = 25m - 1200$</p> <p>$8m = 1200$</p> <p>$m = 150$</p> <p>Thus, the value of $\mathrm{m}$ is equal to:</p> <p>Option C: 150</p>