Propositiones ad Acuendos Juvenes/Problems/13 - De Rege et de Eius Exercitu

by : Problem $13$

 * De Rege et de Eius Exercitu: A King's Army
 * A king ordered his servant to collect an army from $30$ manors,
 * in such a way that from each manor he would take the same number of men as he had collected up till then.
 * The servant went to the first manor alone;
 * to the second he went with one other;
 * to the next he took $3$ with him.


 * How many were collected in all?

Solution

 * $2^{30} = 1 \, 073 \, 741 \, 824$

An extraordinarily large army.

Proof
The assumption is that on visiting the first manor, the servant has collected himself, so to speak.

Otherwise the question never gets started::
 * the servant collects no soldiers from the first manor,
 * the same number as collected so far (none) from the second, and so on,
 * until at the $30$th manor he still has collected no soldiers.

So the assumption as made is that on visiting the $1$st manor, the servant has collected $2^0 = 1$ soldier, namely, himself.

After visiting the first manor, the servant has $2^1 = 2$ soldiers.

Let $T_n$ denote the number of soldiers collected after visiting manor $n$.

On visiting manor $n + 1$, the servant has collected $T_n$ soldiers.

Thus on leaving manor $n + 1$, the servant has collected $T_n + T_n$, that is, $2 T_n$ soldiers.

It follows by Principle of Mathematical Induction that after visiting the $n$th manor, he has a total of $2^n$ soldiers.

The result follows.