Question:hard

On day one,there are 100 particles in a laboratory experiment.On day n,where n≥2,one out of every n particles produces another particle.If the total number of particles in the laboratory experiment increases to 1000 on day m,then m equals

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In basic terms, the number of particles rises by 50 each day. 
From 100 particles on the first day, we must attain 1000 particles. 
Alternatively, 900 more particles are required. 
It takes 18 days for the particle count to rise by 900 at the rate of 50 each day. 
Consequently, there will be 1,000 articles on day 19.
Updated On: Jan 15, 2026

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The Correct Option is A

Solution and Explanation

Given:
- Initial count (Day 1): 100 particles.
- Subsequent days (Day n, where n ≥ 2): 1/n of existing particles reproduce.

Objective: Determine the day (m) when the total particle count reaches 1000.

Analysis:
Day 2 (n=2): 100 + (100/2) = 150 particles.
Day 3 (n=3): 150 + (150/3) = 200 particles.
Day 4 (n=4): 200 + (200/4) = 250 particles.

An increment of 50 particles is observed between Day 2 and Day 3, and between Day 3 and Day 4. This indicates a consistent addition of 50 particles per day after Day 1.

Calculation for Day m:
Total particles = Initial particles + (Incremental particles per day * Number of increments).
The total increase required is 1000 - 100 = 900 particles.
Each increment adds 50 particles.
Number of increments = 900 / 50 = 18.
These 18 increments occur from Day 2 to Day m. Therefore, the number of days for these increments is m - 1.
\(m - 1 = 18\)
\(m = 18 + 1 = 19\)

Conclusion: The total number of particles reaches 1000 on day \(19\).

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