If n is a root of the equation x² (1 − ac) − x (a² + c²) − (1 + ac) = 0 and if n HM’s are inserted between a and c, show that the difference between the first and the last mean is equal to ac(a − c).
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Let n HM’s between a and c are H 1 , H 2 , H 3 , ..... H n
∴ 
, 
, 
, ...... , 
, 
are in A.P.
suppose 'd' is common difference of this A.P.
∴ 
= 
+ (n + 1) d ⇒ d = 
so 
= 
+ 
= 
⇒ H 1 = 
and 
= 
+ 
= 
⇒ H n = 
∴ H 1 – H n = ac(n + 1) 
= ac(n + 1) 
H 1 – H n = 
.......(i)
n is a root of x 2 (1 – ac) – x(a 2 + c 2 ) – (1 + ac) = 0
we have n 2 (1 – ac) – n(a 2 + c 2 ) – (1 + ac) = 0
⇒ n 2 ac + n(a 2 + c 2 ) + ac = (n 2 – 1) .......(ii)
∴ from (i) and (ii), we get
So H 1 – H n = ac(a – c) Ans.
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