Question

X,, X2, X3.... and y, y, yz,... are arithmetic sequences. Prove that all
the points with coordinates in the sequence (x,y),(x,y2), (X3, Y3),...
of number pairs, are on the same line

Answer 1

Question:

Prove that if $x_1,\ x_2,\ x_3,\dots$ and $y_1,\ y_2,\ y_3,\dots$ form two arithmetic sequences, then all points with coordinates $(x_1,\ y_1),\ (x_2,\ y_2),\ (x_3,\ y_3),\dots$ are collinear.

Solution:

Let the common difference of the first AP be $d_x,$ then,

$d_x=x_2-x_1=x_3-x_2=x_4-x_3=\dots$

Generally,

$d_x=x_{n+1}-x_n,\quad n\in\mathbb{N}$

But we can see that,

$x_3-x_1=2d_x=(3-1)d_x\\\\x_5-x_1=4d_x=(5-1)d_x\\\\x_6-x_3=3d_x=(6-3)d_x$

So, generally,

$x_p-x_q=(p-q)d_x,\quad p,\ q\in\mathbb{N}$

Let the common difference of the second AP be $d_y,$ then, similarly,

$y_p-y_q=(p-q)d_y,\quad p,\ q\in\mathbb{N}$

The points with coordinates $(x_1,\ y_1),\ (x_2,\ y_2),\ (x_3,\ y_3),\dots$ are in the form $(x_n,\ y_n)$ for $n\in\mathbb {N}.$ Let me take two arbitrary points $(x_p,\ y_p)$ and $(x_q,\ y_q)$ among them for $p,\ q\in\mathbb {N}.$

Then the slope of the line joining these two points is given by,

$m=\dfrac {y_p-y_q}{x_p-x_q}\\\\\\m=\dfrac {(p-q)d_y}{(p-q)d_x}\\\\\\m=\dfrac {d_y}{d_x}$

which is constant for every possible values of p and q. This proves that the points $(x_1,\ y_1),\ (x_2,\ y_2),\ (x_3,\ y_3),\dots$ are collinear.

QED

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