Years to reach one million

Statement

The headmistress invests the $50000$ euros collected in $2024$ in shares that grow by $18\%$ each year. After how many years will the invested amount have reached one million euros?

Solution

After $n$ years the capital is $50000\cdot 1.18^{\,n}$. Require it to reach one million: $50000\cdot 1.18^{\,n}=1\,000\,000 \iff 1.18^{\,n}=20.$ Taking logarithms: $n=\frac{\ln 20}{\ln 1.18}\approx \frac{2.9957}{0.1655}\approx 18.1.$ Since the capital grows in yearly steps, one million is exceeded in the 19th year.

$\boxed{n=\dfrac{\ln 20}{\ln 1.18}\approx 18.1\ \Rightarrow\ \text{in the 19th year}}$