The father wanted

The father wanted to divide the inheritance equally among all his sons and ordered: the eldest son would receive 1000 CZK and one eighth of the remainder, the second would receive 2000 CZK and one eighth of the new remainder, the third would receive 3000 CZK and one eighth of the remaining remainder, etc. until the entire amount was used up. How many sons were there and what inheritance did they receive?

Final Answer:

d =  49000 Kc
n =  7

Step-by-step explanation:

$$\begin{gathered} 1000+\frac{1}{8}\cdot (d-1000)=2000+\frac{1}{8}\cdot (d-2000-1000-\frac{1}{8}(d-1000)) \\ 1000+1/8\cdot (d-1000)=2000+1/8\cdot (d-2000-1000-1/8\cdot (d-1000)) \\ 0.125d=6125 \\ d=\frac{6125}{0.125}=49000 \\ d=49000=49000\text{Kc} \end{gathered}$$

$$\begin{gathered} n_1=1000+\frac{1}{8}\cdot (d-1000)=1000+\frac{1}{8}\cdot (49000-1000)=7000\text{Kc} \\ n=d/n_1=49000/7000=7 \\ \text{ Verifying Solution: } \\ {a}_1=1000+\frac{d-1000}{8}=1000+\frac{49000-1000}{8}=7000 \\ {d}_1=d-a_1=49000-7000=42000 \\ {a}_2=2000+\frac{d_1-2000}{8}=2000+\frac{42000-2000}{8}=7000 \\ {d}_2=d_1-a_2=42000-7000=35000 \\ {a}_3=3000+\frac{d_2-3000}{8}=3000+\frac{35000-3000}{8}=7000 \\ {d}_3=d_2-a_3=35000-7000=28000 \\ {a}_4=4000+\frac{d_3-4000}{8}=4000+\frac{28000-4000}{8}=7000 \\ {d}_4=d_3-a_4=28000-7000=21000 \\ {a}_5=5000+\frac{d_4-5000}{8}=5000+\frac{21000-5000}{8}=7000 \\ {d}_5=d_4-a_5=21000-7000=14000 \\ {a}_6=6000+\frac{d_5-6000}{8}=6000+\frac{14000-6000}{8}=7000 \\ {d}_6=d_5-a_6=14000-7000=7000 \\ {a}_7=7000+\frac{d_6-7000}{8}=7000+\frac{7000-7000}{8}=7000 \\ {d}_7=d_6-a_7=7000-7000=0 \\ D=a_1+a_2+a_3+a_4+a_5+a_6+a_7=7000+7000+7000+7000+7000+7000+7000=49000 \end{gathered}$$

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