Question

How can you write an equation to represent this problem for any number of guards?

A man entered an orchard through 7 gates and picked some apples. When he left, he gave the first guard half his apples and 1 apple more. To the second guard he gave half his remaining apples and 1 more. He did the same to each of the remaining five guards and left the orchard with 1 apple. How many apples did he gather in all?

Answer 1

Explanation:

This problem can be represented by the equation: x + (x/2 - 1) + (x/4 - 1) + (x/8 - 1) + ... + (x/128 - 1) + 1 = x, where x is the total number of apples gathered.

This equation represents the process described in the problem, where the man gives half of his remaining apples (x/2, x/4, x/8, etc.) to each guard, minus 1 apple each time, until he is left with 1 apple.