Recursive Sequence Growth

Let $x_n$ be the number of organisms on the $n^{\text{th}}$ day. Given $x_1 = 2$ and $x_n = 2 x_{n-1} + 3$.

Add 3 to both sides: $x_n + 3 = 2 x_{n-1} + 6 = 2(x_{n-1} + 3)$ Let $y_n = x_n + 3$. Then $y_n = 2 y_{n-1}$ with $y_1 = x_1 + 3 = 5$. So $y_n = 5 \times 2^{n-1}$, which means $x_n = 5 \times 2^{n-1} - 3$.

We want $x_n > 10^6$: $5 \times 2^{n-1} - 3 > 10^6 \implies 5 \times 2^{n-1} > 1000003 \implies 2^{n-1} > 200000.6$

Powers of 2:

• $2^{17} = 131072$
• $2^{18} = 262144 > 200000.6$

Thus, $n - 1 = 18 \implies n = 19$.