Question

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

Answer 1

If the line which contains the points (x₁, y₁), (x₃, y₃) also contains (x, y), then

\(\frac{y-y_1}{x-x_1}=\frac{y_3-y_1}{x_3-x_1}=\frac{2d}{2d}=1\)

y - y₁ = 1

y = 1 + y₁

x - x₁ = 1

x = 1 + x₁

The point (x, y) is one greater than (x₁,y₁)

Therefore, (x₁, y₁), (x₂, y₂) …………. will be in the same line