A man entered an apple orchard through 7 gates. After picking some apples, he left. Upon leaving, he was required to give 1/2 of his apples plus an apple to the guard at the first gate. When he got to the second gate he was required to give half of his remaining apples plus an apple to that guard. This continued for each of the remaining 5 gates. When he finally got out of the last gate he had only one apple for himself. How many apples did he pick?
11.5 APPLES
To solve this problem, let's work backwards. We know that when the man left the last gate, he had only one apple remaining. From there, we can trace back the process of giving away apples at each gate.
At the last gate, the man had 1 apple for himself before giving away half of his remaining apples plus one apple to that guard. So, we can say that after leaving the last gate, he had:
1 apple (for himself) + 1 apple (given to the last guard) = 2 apples
Now, let's consider the second-to-last gate. We know that after leaving this gate, he had 2 apples. Again, let's calculate how many apples he had before giving away half of his remaining apples plus one apple to the guard at this gate:
2 apples (for himself) + 1 apple (given to the second-to-last guard) = 3 apples
We can continue this process for each gate until we reach the first gate. Let's summarize the results:
After leaving the third gate: 3 apples (for himself) + 1 apple (given to the third guard) = 4 apples
After leaving the fourth gate: 4 apples (for himself) + 1 apple (given to the fourth guard) = 5 apples
After leaving the fifth gate: 5 apples (for himself) + 1 apple (given to the fifth guard) = 6 apples
After leaving the sixth gate: 6 apples (for himself) + 1 apple (given to the sixth guard) = 7 apples
After leaving the seventh gate: 7 apples (for himself) + 1 apple (given to the seventh guard) = 8 apples
Therefore, the man initially had 8 apples before entering the first gate of the apple orchard.