Solve the following problem from Fibonacci\'s Liber abaci: A man left to his eld

Question

Solve the following problem from Fibonacci's Liber abaci: A man left to his eldest son one bezant and a seventh of what was left; then from the remainder, to his next son he left two bezants and a seventh of what was left; then, from the new remainder, to his third son he left three bezants and a seventh of what was left. He continued this way, giving each son one bezant more than the previous one and a seventh of what remained. By this division it developed that the last son received all that was left and that all the sons shared equally. How many sons were there and how large was the man's estate?

Explanation / Answer

Consider ES as the size of the estate , and N denote the number of sons, and ESi is the amount son number i receives.

Consider the formula ,

ES1= 1 + [(ES-1) /7]   

And , ES2 = 2 + [(ES-ES1-2)/7]

On equating these two formula's we get ES1 = 6 . This will represent that the same amount is given to each of the son .

Now by substituting 6 is the first equation , we get

ES1= 1 + [(ES-1) /7]   

6 = 1 + [(ES-1) /7] = 36

ES = 6N , Therefore there are 6 number of sons and 36 bezants of estate.

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