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Mathematics Question on Sequence and series

If log⁡(x+z)+log⁡(x−2y+z)=2log⁡(x−z),\log (x + z) + \log (x - 2y + z) = 2 \log (x - z), then x,y,z x, y, z are in

A

A.P

B

H.P

C

G.P

D

None of these

Answer

H.P

Explanation

Solution

log⁡(x+z)+log⁡(x−2y+z)=2log⁡(x−z)\log (x + z) + \log (x - 2y + z) = 2 \log (x - z)
⇒   log⁡(x+z)?(x−2y+z)=log⁡(x−z)2\Rightarrow \:\:\: \log (x + z)?(x - 2y + z) = \log (x - z)^2
⇒   (x+z)2−2y(x+z)=(x−z)2\Rightarrow \:\:\: (x + z)^2 -2y (x + z) = (x-z)^2
⇒   x2+z2+2xz−2yx−2yz=x2+z2−2zx\Rightarrow \:\:\:x^2 + z^2 + 2xz - 2yx - 2yz = x^2 + z^2 - 2zx
⇒   xz−yx−yz=−zx ⇒ 2zx=xy+yz\Rightarrow \:\:\:xz - yx - yz = -zx \: \Rightarrow \: 2zx = xy + yz
2y=1x+1z⇒1x,1y,1z\frac{2}{y}=\frac{1}{x}+\frac{1}{z} \Rightarrow \frac{1}{x} , \frac{1}{y} , \frac{1}{z} are in A.P.
∴    x,y,z\therefore \:\:\:\: x,y,z are in H.P.