2021 JMC 10 Problems/Problem 14

Problem

For a certain $b,$ the base $b$ numbers $24_b,n,57_b,72_b, \ldots$ form an increasing arithmetic sequence in that specific order. Then, what is the value of $n,$ expressed in base $10?$

$\textbf{(A) } 47 \qquad \textbf{(B) } 51 \qquad \textbf{(C) } 63 \qquad \textbf{(D) } 64 \qquad \textbf{(E) } 75$

Solution

Note that $24_b = 2b+4,57_b = 5b+7,$ and $72_b = 7b+2.$ Because they are terms in an arithmetic sequence, the difference between $57_b$ and $24_b$ must be twice the difference between $72_b$ and $57_b.$ It follows that $(5b+7)-(2b+4) = 2((7b+2) - (5b+7)) \implies b = 13.$ So the terms expressed in base $10$ are $2\cdot13+4, n , 5\cdot 13+7, 7\cdot 13+2, \dots$ and $n = \tfrac{30+72}{2} = 51.$

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2021 JMC 10 Problems/Problem 14

Problem

Solution