The father wanted

A father wanted to divide his inheritance equally among all his sons and ordered: the eldest son would receive 1,000 CZK and one eighth of the remainder; the second would receive 2,000 CZK and one eighth of the new remainder; the third would receive 3,000 CZK and one eighth of the remaining amount; and so on until the entire amount was used up. How many sons were there, and what was the total inheritance?

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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