2006/07 AHSMC Part 1 Problem 12

Problem 12 of 16

All positive integers which can be expressed as a sum of one or more different integer powers of $5$ are written in increasing order. The first three terms of this sequence are $1$ , $5$ and $6$ . The fiftieth term is

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Solution

We change the base-ten number $50$ to base-two and obtain the number $110010$ , representing $2^5 + 2^4 + 2^2$ . We now interpret this as a base-five number, representing $5^5 + 5^4 + 5^2$ . Changing this number to base-ten, we obtain $3755$ . The answer is (b).

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