Suppose a, b, c are in A.P. and a 2 , b 2 , c 2 are in G.P. If a < b < c and a + b + c =3/2, then the value of a is -
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Since a, b, c are in A.P.
∴ b = a + d, c = a + 2d,
where d is a common difference, d > 0
Again since a 2 , b 2 , c 2 are in G.P.
∴ a 2 , (a + d) 2 , (a + 2d) 2 are in G.P.
⇒ (a + d) 4 = a 2 (a + 2d) 2
Or (a + d) 2 = ± a (a + 2d)
⇒ a 2 + d 2 + 2ad = ± (a 2 + 2ad)
Taking (+) sign, d = 0 (not possible as a < b < c)
Taking (–) sign,
2a 2 + 4ad + d 2 = 0
⇒ 2a 2 + 4a 
= 0
⇒ 4a 2 – 4a – 1 = 0
∴ a = 
± 
.
Here d = 
– a > 0. So, a < 
Hence, a = 
– 
.
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