Question

A man entered an orchard through 7 gates, and there took a certain number of apples.

When he left the orchard, he gave the first guard half of the apples that he had and 1

apple more. To the guard at the second gate, he gave half of his remaining apples and 1

apple more. He did the same with each of the remaining 5 guards and left the orchard

with 1 apple. How many apples did he gather in the orchard?

Answer 1

Before 7th gates a man has $(1+1)\cdot2=4$ apples.

Before 6th gates a man has $(4+1)\cdot2=10$ apples.

Before 5th gates a man has $(10+1)\cdot2=22$ apples.

Before 4th gates a man has $(22+1)\cdot2=46$ apples.

Before 3rd gates a man has $(46+1)\cdot2=94$ apples.

Before 2nd gates a man has $(94+1)\cdot2=190$ apples.

Before 1st gates a man has $(190+1)\cdot2=382$ apples.

A man gathered 382 apples in the orchard.