Find all the numbers $x$ and $y$ such that $x, x + 2 y, 2 x + y$ form an $A . P$ . while the numbers ${(y + 1)}^{2}, x y + 5, {(x + 1)}^{2}$ form a $G . P$ . Write down the progressions.

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Solution

The condition for $A . P . \Rightarrow x = 3 y$
The condition for $G . P .$
$\Rightarrow {(x y + 5)}^{2} = {(x + 1)}^{2} {(y + 1)}^{2}$
$\Rightarrow {(3 y^{2} + 5)}^{2} - {[(3 y + 1) (y + 1)]}^{2} = 0,$
$\Rightarrow (y - 1) (3 y^{2} + 2 y + 3) = 0,$
$∴ y = 1, - 1 + 2 \sqrt{2 i}, - 1 - 2 \sqrt{2 i}$
$∴ y = 1$ $x = 3$
Ans.(i)
$A . P .3, 5, 7$ , $G . P .4, 8, 16$ .
(ii)
$A . P . - 1 + 2 \sqrt{2 i}, \frac{- 5 + 10 \sqrt{2 i}}{3}, \frac{- 7 + 14 \sqrt{2 i}}{3}$
$G . P . \frac{- 4 + 8 \sqrt{2 i}}{9}, \frac{8 - 4 \sqrt{2 i}}{3}, - 8$
(iii)
$A . P . - 1 - 2 \sqrt{2 i}, \frac{- 5 - 10 \sqrt{2 i}}{3}, \frac{- 7 - 14 \sqrt{2 i}}{3}$
$G . P . \frac{- 4 - 8 \sqrt{2 i}}{9}, \frac{8 + 4 \sqrt{2 i}}{3}, - 8$
Note : Answer $(i i)$ and $(i i i)$ are conjugates.

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