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If x+y+z=15 when a,x,y,z,b are in AP and 1/x+1/y+1/z=5/3 when a,x,y,z,b are in HP then
a.GM of a & b is 3
b.AM of a & b is 6
c.1 possible value of a+2b is 11
d.greatest value of a-b is 8
[Ans: (a),(c),(d)]
for AP, let x=a+d, y=a+2d, z=a+3d.
x+y+z=15
or, 3a+6d=15
or, a+2d=5 or, y=5
but y=b-2d
2y=10
a+b=10 ...(i)
for HP, let 1/x = 1/a + f , 1/y = 1/a + 2f , 1/z = 1/a + 3f
1/x + 1/y + 1/z = 5/3
or, 3/a + 6f = 5/3 or, 1/a + 2f = 5/9.
or, 1/y =5/9 or, 1/b - 2f = 5/9.
thus, 1/a + 1/b = 10/9 ....(ii)
from (i) and (ii)
ab = 9 or, (ab)1/2 = 3
solve (i) and (ii) to get other two options.
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