Question

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

• A. 19
• B. 16
• C. 17
• D. 18

Hint:

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.

Answer 1

The Correct Option is A

Given:
- Day one: 100 particles 
- Day n (where n ≥ 2): One out of every n particles produces another particle.

We want to find the value of m when the total number of particles reaches 1000 on day m.

Step 1: Day Two
On day two (n = 2), half of the 100 particles produce another particle.
So, the total becomes 100 + 50 = 150 particles.

Step 2: Day Three
On day three (n = 3), one-third of the 150 particles produce another particle.
So, the total becomes 150 + 50 = 200 particles.

Step 3: Day Four
On day four (n = 4), one-fourth of the 200 particles produce another particle.
So, the total becomes 200 + 50 = 250 particles.

We can observe a pattern here: in each step, 50 particles are added.

Step 4: Day m
On day m, the total becomes 1000 particles.
From day 1 to day m, 50 particles are added in each step after day 1.

So, the number of steps from day 1 to day m is:
\(m - 1 = \frac{1000 - 100}{50}\)
\(m - 1 = \frac{900}{50} = 18\)
\(m = 18 + 1 = 19\)

Therefore, the value of m is \(19\).