4. 

On a coordinate line, 16 points are marked and numbered from left to right. The coordinate of any point, except for the extreme points, is equal to half the sum of the coordinates of the two adjacent points. Find the coordinate of the fifth point if the first point has a coordinate of 2 and the sixteenth point has a coordinate of 47.

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# Solution

Solution. Let $a, b$ and $c$ be the coordinates of three consecutive points (from left to right). Then $b=\frac{a+c}{2}$, which means the second point is the midpoint of the segment with endpoints at the neighboring points. This condition holds for any triple of consecutive points, meaning the distances between any neighboring points are the same. The distance between the extreme points is 47-2=45, and there are 15 equal intervals between them, so the distance between neighboring points is $45: 15=3$. There are 4 equal intervals of length 3 between the fifth and the first point, so the coordinate of the fifth point is $2+3 \cdot$ $4=14$.

Answer: 14