AMC 12A 2023 (Problem 17)

Flora the frog starts at

0

on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance

m

with probability

\frac{1}{2^m}

. What is the probability that Flora will eventually land at

10

\text{(A)}\;\frac{5}{512}

\text{(B)}\;\frac{45}{1024}

\text{(C)}\;\frac{127}{1024}

\text{(D)}\;\frac{511}{1024}

\text{(E)}\;\frac{1}{2}