Three-four-five in Pascal's triangle

MediumCombinatorics Brainteasers~6m

Some row of Pascal's triangle contains three consecutive entries in the ratio $3 : 4 : 5$ . What is the smallest row index $n$ for which this happens? (Row $n$ has entries $\binom{n}{0}, \dots, \binom{n}{n}$ .)

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Consecutive-entry ratio: $\binom{n}{k+1}/\binom{n}{k} = (n-k)/(k+1)$ . Set the two ratios to $4/3$ and $5/4$ .
That gives $3n=7k+4$ and $4n=9k+14$ . Solve for $k$ , then $n$ .

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Three-four-five in Pascal's triangle

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