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Doubling stakes

Mental mathMedium

A player stakes $1$ on the first round and doubles the stake every round, up to a stake of $256$ on the last round. What is the total staked over all rounds?

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Hint

Compare the sum with the next term in the pattern.

Solution

The stakes are $1, 2, 4, \dots, 256$ , powers of $2$ up to $2^{8}$ .
Each stake is $1$ more than the sum of all previous stakes, so the sum of the series is one less than the next term, $512$ .
Check with a small case: $1 + 2 + 4 = 7 = 8 - 1$ .
So the total is $512 - 1 = 511$ .

Answer: $511$

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