Aman picks apples. on his way to town, he has to pass seven gates. at each gate stands a guardian. claiming half of the apples and one apple extra. at the end, the man has only one apple left. how many apples did he have at the beginning?

The man is left with one apple. Before the seventh gate, he had (1+1)x2, or 4 apples. Before the sixth gate, he had (4+1)x2, or 10 apples. Before that, he had (10+1)x2, or 22 apples. Thus, the number of apples he had is as follows:
1,4,10,22,46,94,190,382. So, he started with 382 apples
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