Several points were marked on the straight line. After that, another point was marked between

Several points were marked on the straight line. After that, another point was marked between each two adjacent points. This seal was repeated twice more (3 times in total). As a result, 113 points were marked on the straight line. How many points were originally marked?

Suppose that x points were initially marked.

After the first transformation, one appears between each adjacent points. This means that there are one less new points than it was originally. Total number of points:

x + (x – 1) = 2 * x – 1

The second transformation follows the same scheme, adding ((2 * x – 1) – 1) new points.

(2 * x – 1) + ((2 * x – 1) – 1) = 4 * x – 3

The third transformation is similar:

(4 * x – 3) + ((4 * x – 3) – 1) = 8 * x – 7.

8 * x – 7 = 113;

8 * x = 120;

x = 15.

Answer: initially there were 15 points.