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Question

1x+y,12y,1y+z are in A.P. Prove that x, y, z are the consecutive terms of G.P

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Solution

Given 1x+y,12y,1y+z are in A.P.
To prove that x, y, z are the consecutive terms of G.P, then we have
Since, 1x+y,12y,1y+z are in A.P.
then, 12y=1x+y+11y+z2
Also we can calculate the common difference
12y1x+y=dxyx+y=2yd(1)
1y+z12y=xyx+y=2yd(2)
from (1) and (2) we get xyx+y=yzy+z
(xy)(y+z)=(x+y)(yz)
xy+xzy2yzxyxz+y2yz2y22xz=02(y2xz)=0y2xz=0y2=xz
Hence the terms x, y, z are the consecutive terms of a G.P

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