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
12y−1x+y=d⇒x−yx+y=2yd−−−(1)
1y+z−12y=⇒x−yx+y=2yd−−−(2)
from (1) and (2) we get x−yx+y=y−zy+z
⇒(x−y)(y+z)=(x+y)(y−z)
⇒xy+xz−y2−yz−xy−xz+y2−yz⇒2y2−2xz=0⇒2(y2−xz)=0⇒y2−xz=0⇒y2=xz
Hence the terms x, y, z are the consecutive terms of a G.P