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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_17 / Flora the frog starts at 0 on the number line and makes a sequence of jumps to…

Problem

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problem

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? Write the answer as a simplified fraction

\frac{m}{n}

, find

m+n

Plain-text mathematical notation (without MathML)

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 (1)/(2^(m)).
What is the probability that Flora will eventually land at 10? Write the answer as a simplified fraction (m)/(n), find m+n

Original LaTeX notation

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? Write the answer as a simplified fraction $\frac{m}{n}$, find $m+n$

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_17
task_id: 7165f327-8647-550a-a28a-90c2e4f4e770
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2023~5fAMC~5f12A~5fProblems~2fProblem~5f17
task_revision_id: 2
config / split: default / train

Official source

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