Question:
if a,b and c are in HP , then a-b/2 , b/2 , c-b/2 are in ?
Answer:
Given, a, b, c are in HP
1/b = 1/a + 1/c
=> 1/b = (a + c)/ac
=> ac = b(a + c)
=> ac = ab + bc
=> ac - ab - bc = 0
=> ac - ab - bc + b2 = b2 {adding b2 on both side}
=> a(c - b) - b(c - b) = b2
=> (a - b)*(c - b) = b2
=> (a - b)/2 *(c - b)/2 = b2 /4
=> (a - b)/2 *(c - b)/2 = (b/2)2
=> (a - b)/2 , b/2 , (c - b)/2 are in GP
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