JEE Main 2024 — Sequences And Series Question with Solution

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:

Initially, there are $$\mathrm{m}$$ computers, and it is estimated that with these $$\mathrm{m}$$ computers, the assignment can be completed in 17 days.

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.

To begin with, let's define the total work (W) in terms of the number of computers and days:

The total work (W) is given by:

The amount of work completed each day with $$\mathrm{m}$$ computers for 17 days:

$$W=17m$$

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.

This can be formulated as:

Total work done over 25 days with decrement in the number of computers:

$$W=m+(m-4)+(m-8)+\dots +\left[m-4\times (n-1)\right]$$

where $$n$$ is the number of days. Here, $$n=25$$.

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:

$$S_n=\frac{n}{2}\left[2a+(n-1)d\right]$$

Plugging in the values:

$$S_{25}=\frac{25}{2}\left[2m+(25-1)(-4)\right]$$

$$S_{25}=\frac{25}{2}\left[2m-96\right]$$

$$S_{25}=\frac{25}{2}\left[2m-96\right]=25(m-48)$$

This work should be equivalent to the work calculated earlier, so:

$$17m=25(m-48)$$

Solving for $$\mathrm{m}$$:

$$17m=25m-1200$$

$$8m=1200$$

$$m=150$$

Thus, the value of $$\mathrm{m}$$ is equal to:

Option C: 150