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