I can see why the following formula is correct, but I'm not sure how to set about proving it.
A man needs to spread 4000kg of sand over his garden. He decides to spread 200kg every day, but after the first day he discovers that he can only spread 95% of the amount he spread the previous day.
Show that after $n$ days, he has spread $4000(1 - 0.95^n)$.
Let $(u_n)_{n\geq 1}$ the amount of sand (in kg) he spread the nth day. Now, "he decides to spread 200kg everyday" gives $u_n = 200$ for $n\geq 1$. But "after the first day discovers that he can only spread 95% of the amount he spread the previous day", so actually $u_n = 0.95 u_{n-1}$ for $n \geq 2$. This shows that the sequence $(u_n)$ is geometric with common ratio $r=0.95$, thus $$ u_n = u_1 r^{n-1} = 200\times 0.95^{n-1}. $$ After $n$ days, the amount of sand he has spread is $$ \begin{align*} u_1 + u_2 + \dots + u_n &= 200\times (1 + 0.95 + \dots + 0.95^{n-1}) \\ &= 200\times \frac{1-0.95^n}{1-0.95} \\ &= 4000(1-0.95^n). \end{align*} $$
