IMC 2012 Q17
17. The first term of a sequence of positive integers is 6. The other terms in the sequence follow these rules:
if a term is even then divide it by 2 to obtain the next term;
if a term is odd then multiply it by 5 and subtract 1 to obtain the next term.
For which values of \(n\) is the \(n\)th term equal to \(n\)?
- A 10 only
- B 13 only
- C 16 only
- D 10 and 13 only
- E 13 and 16 only
Intermediate Mathematical Challenge 2012, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 13 and 16 only
The terms of the sequence are 6, 3, 14, 7, 34, 17, 84, 42, 21, 104, 52, 26, 13, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, … . As can be seen, there will now be no other terms in the sequence other than 4, 2 and 1. It can also be seen that the only values of \(n\) for which the \(n\) th term = \(n\) are 13 and 16.
Solution © UK Mathematics Trust, from the IMC 2012 solutions. Longer versions with investigations: Solutions and investigations (PDF).