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a new addition to the frog family

Source: 2023 AMC12A #17

November 9, 2023
probabilityAMCAMC 122023 AMC2023 AMC 12A

Problem Statement

Flora the frog starts at 00 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 mm with probability 12m\frac{1}{2^m}. What is the probability that Flora will eventually land at 1010?
<spanclass=′latex−bold′>(A)</span>5512<spanclass=′latex−bold′>(B)</span>451024<spanclass=′latex−bold′>(C)</span>1271024<spanclass=′latex−bold′>(D)</span>5111024<spanclass=′latex−bold′>(E)</span>12<span class='latex-bold'>(A) </span> \frac{5}{512} \qquad <span class='latex-bold'>(B) </span> \frac{45}{1024} \qquad <span class='latex-bold'>(C) </span> \frac{127}{1024} \qquad <span class='latex-bold'>(D) </span> \frac{511}{1024} \qquad <span class='latex-bold'>(E) </span> \frac{1}{2}
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