If (x , y , z) are in A.P. and and (\tan ^ { - 1 } z) are al... | MATH - ARITHMETIC PROGRESSION | SolveeIt

If $x , y , z$ are in A.P. and and $\tan ^ { - 1 } z$ are also in A.P., then.

$x = y = z$

$x = y = - z$

$x = 1 ; y = 2 ; z = 3$

$x = 2 ; y = 4 ; z = 6$ (5) $x = 2 y = 3 z$

Answer

$x = y = z$

Explanation

$2 \tan ^ { - 1 } y = \tan ^ { - 1 } x + \tan ^ { - 1 }$ z

⇒ $\tan ^ { - 1 } \left( \frac { 2 y } { 1 - y ^ { 2 } } \right) = \tan ^ { - 1 } \left( \frac { x + z } { 1 - x z } \right)$

⇒ $\frac { 2 y } { 1 - y ^ { 2 } } = \frac { x + z } { 1 - x z }$

But $2 y = x + z$

∴ $1 - y ^ { 2 } = 1 - x z$ ⇒ $y ^ { 2 } = x z$

$\because x y z$ are both in G.P. and A.P.,

Question: If \(x , y , z\) are in A.P. and <img src="https://cdn.pureessence.tech/canvas_632.png?top_left_x=30...