The king

The king divided ducats among his sons. He gave the eldest son a certain number of ducats, gave the younger one ducat less, gave the other one ducat less, and proceeded to the youngest. Then he returned to his eldest son, gave him one ducat less than a while ago to the youngest, and distributed it in the same way as in the first round. In this round, one ducat came out for his youngest son. The eldest son received a total of 21 ducats.
Find how many sons the king had and how many ducats he gave them.

Final Answer:

n =  7
s =  105

Step-by-step explanation:

$$\begin{gathered} n+2n=21 \\ n+2\cdot n=21 \\ 3n=21 \\ n=\frac{21}{3}=7 \\ n=7 \end{gathered}$$

$$\begin{gathered} a_1=7+14=21 \\ {a}_2=6+13=19 \\ {a}_3=5+12=17 \\ {a}_4=4+11=15 \\ {a}_5=3+10=13 \\ {a}_6=2+9=11 \\ {a}_7=1+8=9 \\ s=a_1+a_2+a_3+a_4+a_5+a_6+a_7=21+19+17+15+13+11+9=105 \end{gathered}$$

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