Pages MO Z6-I-4

Once the king called all his pages and placed them in a line. To the first page he gave a certain number of ducats, to the second he gave two ducats less, to the third again two ducats less, and so on. When he reached the last page, he gave him the appropriate number of ducats, turned around, and in the same way proceeded to the beginning of the line (i.e. to the penultimate page he gave two ducats less than a moment ago to the last one, etc.). The first page in this round got two ducats. Then one of the pages found out that he has 32 ducats. How many pages could the king have had, and how many ducats in total could he have distributed to them?

Determine all possibilities.

Final Answer:

n₁ =  8
d₁ =  240
n₂ =  16
d₂ =  992

Step-by-step explanation:

$$\begin{gathered} d(n)=(n-1)\cdot 4\cdot n+2\cdot n \\ {n}_1=8 \end{gathered}$$

$d_1=(n_1-1)\cdot 32+16=(8-1)\cdot 32+16=240$

$n_2=2\cdot n_1=2\cdot 8=16$

$d_2=(n_2-1)\cdot 64+32=(16-1)\cdot 64+32=992$

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