Let $$A_{1},A_{2},.....A_{n}$$ be then points on the straight - line y = px + q. The coordinates of $$A_{k}is(X_{k},Y_{k})$$, where k = 1, 2, ...n such that $$X_{1},X_{2}....X_{n}$$ are in arithmetic progression. The coordinates of $$A_{2}$$ is (2,–2) and $$A_{24}$$ is (68, 31).
$$x_1 , x_2 , x_3,......., x_n$$ are in A.P. Let the first term be $$a$$ and common difference be $$d$$
Also, $$x_2 = a + d = 2$$
and $$x_{24} = a + 23d = 68$$
Solving above equations, we get : $$a = -1$$ and $$d = 3$$
=> $$x_8 = a + 7d = -1 + 7(3) = 20$$
To find y coordinate, we will use $$y = px + q$$
$$\because A_2 (2, -2)$$ and $$A_{24} (68, 31)$$
Substituting in above equation, => $$-2 = 2p + q$$
and $$31 = 68p + q$$
Solving above equations, we get : $$p = \frac{1}{2}$$ and $$q = -3$$
$$\therefore y_8 = p x_8 + q = \frac{1}{2} (20) + (-3) = 7$$
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