Q3 of 23 Page 226

x1, x2, x3,… and y1, y2, y3,… are arithmetic sequences. Prove that all points with coordinates in the sequence (x1, y1), (x2, y2), (x3, y3),… are on the same line.

Let the common difference between the x - coordinate be dx , between the y - coordinate be dy


The three points can be written in terms of their common difference as (x1, y1), (x1 + dx, y1 + dy), (x1 + 2dx, y1 + 2dy)


Slope of first two points


⇒ Slope of first two points


Slope of last two points


⇒ Slope of last two points


Since the slope of first two points and last two points are same hence they are on the same line.


Hence proved


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