Problem 16: A Duel for Jewels

In his will, a King left his collection of jewels to his daughters with the instructions to divide them as follows:

• The eldest daughter should take 1 jewel plus ¹/₇ of whatever is left.
• Next, the second eldest daughter should take 2 jewels plus ¹/₇ of whatever is left.
• The next eldest daughter should then take 3 jewels plus ¹/₇ of whatever is left.
• And so on in the same manner, until the youngest daughter receives whatever is left after all her sisters have taken their jewels.

At first, the youngest daughter thought this was unfair, but the old King was smart: in the end, all the daughters received the same number of jewels.

How many jewels are in the collection, and how many does each daughter receive?

Bonus

What if the fraction was ¹/₈ instead of ¹/₇?

What if the fraction was ¹/_n?