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Question:

On day one, there are 100 particles in a laboratory experiment. On day n, where n greater than or 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.

Explanation:

On nth day ‘n’ particles produce 1 extra particle.
⇒ For every n particles on previous day, their will be (n + 1) particles next day.

∴ On nth day, the number of particles will become $\frac{n + 1}{n}$ times the number of particles of previous day.

⇒ Number of particles after day 2 = 100 × $(\frac{3}{2})$
⇒ Number of particles after day 3 = 100 × $(\frac{3}{2})$ × $(\frac{4}{3})$
⇒ Number of particles after day 4 = 100 × $(\frac{3}{2})$ × $(\frac{4}{3})$ × $(\frac{5}{4})$

...

⇒ Number of particles after day m = 100 × $(\frac{3}{2})$ × $(\frac{4}{3})$ × $(\frac{5}{4})$ × ... × $(\frac{m + 1}{m})$ = 100 × $(\frac{m + 1}{2})$

⇒ 100 × $(\frac{m + 1}{2})$ = 1000

⇒ $(\frac{m + 1}{2})$ = 10

⇒ m = 19

Hence, option (c).

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