23. Several points were marked on a line. Renard then marked another point between each pair of adjacent points on the line. He performed this process a total of four times. There were then 225 points marked on the line. How many points were marked on the line initially?
A 15
B 16
C 20
D 25
E 30
SOLUTION
A

Let the original number of points be $n$. Marking an extra point between each pair of adjacent points would add an extra $n-1$ points, giving $2 n-1$ points in total after applying the process once. When this process is repeated, there would be $2(2 n-1)-1=4 n-3$ points marked after the second application, $2(4 n-3)-1=8 n-7$ points after the third application and $2(8 n-7)-1=16 n-15$ points after the fourth application. The question tells us that there were 225 points after the fourth application of the process and hence $16 n-15=225$, which has solution $n=15$. Therefore there were 15 points marked on the line initially.