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Problem 355

Number theory Difficulty 2.5 Multiple choice CEMC Fermat · Canada · 2023

A sequence has 101 terms, each of which is a positive integer. If
a term, $n$ , is even, the next term
is equal to $\frac{1}{2}n+1$ . If a
term, $n$ , is odd, the next term is
equal to $\frac{1}{2}(n+1)$ . For
example, if the first term is $7$ ,
then the second term is 4 and the third term is 3. If the first term is
$16$ , the 101st term is

Pick one

Official solution

The 1st term is 16.

Since 16 is even, the 2nd term is $\frac{1}{2}\cdot 16 + 1 = 9$ .

Since 9 is odd, the 3rd term is $\frac{1}{2}(9 + 1) = 5$ .

Since 5 is odd, the 4th term is $\frac{1}{2}(5 + 1) = 3$ .

Since 3 is odd, the 5th term is $\frac{1}{2}(3 + 1) = 2$ .

Since 2 is even, the 6th term is $\frac{1}{2}\cdot 2 + 1 = 2$ .

This previous step shows us that when one term is 2, the next term
will also be 2.

Thus, the remaining terms in this sequence are all 2.

In particular, the 101st term is 2.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.