IMC 2015 Q14
14. In a sequence, each term after the first two terms is the mean of all the terms which come before that term. The first term is 8 and the tenth term is 26. What is the second term?
- A 17
- B 18
- C 44
- D 52
- E 68
Intermediate Mathematical Challenge 2015, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 44
Suppose the first three terms of the sequence are \(a\), \(b\), \(c\). Then \(c = \tfrac{1}{2}(a + b)\) and so \(a + b = 2c\). The mean of the first three terms is then \(\tfrac{1}{3}(a + b + c) = \tfrac{1}{3}(2c + c) = c\), so the fourth term is \(c\). Similarly, the following terms are all equal to \(c\). Since one of these terms is 26 and \(a = 8\) then \(b = 2c - a = 52 - 8 = 44\).
Solution © UK Mathematics Trust, from the IMC 2015 solutions. Longer versions with investigations: Solutions and investigations (PDF).