\section*{Problem 13}

We place labeled points on a circle as follows. At step 1, take two points at opposite ends of a diameter and label them both 1 . At step \(n>1\), place a point at the midpoint of each arc created at step n-1 and label it with the sum of the labels at the two adjacent points. What is the total sum of the labels after step \(n\) ?

For example, after step 4 we have: \(1,4,3,5,2,5,3,4,1,4,3,5,2,5,3,4\).


\section*{Solution}

Answer: \(2.3^{\mathrm{n}-1}\).

True for \(\mathrm{n}=1\). The new points added at step \(\mathrm{n}+1\) have twice the sum of the points after step \(\mathrm{n}\), because each old point contributes to two new points. hence the total after step \(n+1\) is three times the total after step \(n\).

