On day one, there are 100 particles in a laboratory experiment. On day $$ n $$, where $$ n \geq 2 $$, 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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To solve this problem:

1. **Initial Observation**:  
   On day 1, there are 100 particles. From day 2 onward, the number of particles increases according to the rule:  
   - On day $$ n $$, one out of every $$ n $$ particles produces an additional particle.

2. **Growth Pattern**:  
   The increase in particles each day is calculated as:  
   $$
   	ext{Increase on day } n = \frac{	ext{Total particles on day } (n-1)}{n}.
   $$
   From the table:  
   - Day 1: Total = 100  
   - Day 2: Increase = $$ \frac{100}{2} = 50 $$; Total = 150  
   - Day 3: Increase = $$ \frac{150}{3} = 50 $$; Total = 200  
   - Day 4: Increase = $$ \frac{200}{4} = 50 $$; Total = 250

The pattern shows that the number of particles increases by **50 every day**.

3. **Calculate Days to Reach 1000 Particles**:  
   Starting with 100 particles, we need an additional:  
   $$
   	ext{Required Increase} = 1000 - 100 = 900.
   $$
   At a rate of **50 particles per day**, it will take:  
   $$
   \frac{900}{50} = 18 	ext{ days}.
   $$
   Adding the first day (day 1), the total number of days is:  
   $$
   m = 18 + 1 = 19.
   $$

Thus, the total number of days required is **19**.

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