Particle Growth Sequence

Let $P_n$ be the number of particles on day $n$. On day 2, 1 out of every 2 particles produces another, so $P_2 = P_1 + \frac{P_1}{2} = P_1 \left(1 + \frac{1}{2}\right) = P_1 \cdot \frac{3}{2}$. On day $n$, $P_n = P_{n-1} \left(1 + \frac{1}{n}\right) = P_{n-1} \cdot \frac{n+1}{n}$. Multiplying from day 2 to day $m$: $P_m = P_1 \times \frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \dots \times \frac{m+1}{m} = P_1 \times \frac{m+1}{2}$. Given $P_1 = 100$ and $P_m = 1000$: $1000 = 100 \times \frac{m+1}{2} \implies 10 = \frac{m+1}{2} \implies m+1 = 20 \implies m = 19$.