If a, b, c an d are in G.P. Show that: (a2 + b2 + c2)(b2 + c2 + d 2) = (ab + bc + cd)2
Solution:
If a, b, c and d are in G.P.
Therefore,
bc = ad ....(1)
b2 = ac ....(2)
c2 = bd ....(3)
We need to prove that, (a2 + b2 + c2)(b2 + c2 + d 2) = (ab + bc + cd)2
Since,
RHS = (ab + bc + cd)2
= (ab + ad + cd)2 [Using (1)]
= [ab + d (a + c)]2
= a2b2 + 2abd (a + c) + d 2 (a + c)2
= a2b2 + 2a2bd + 2acbd + d 2 (a2 + 2ac + c2 )
= a2b2 + 2a2c2 + 2b2c2 + d 2a2 + 2d 2b2 + d 2c2 [Using (1) and (2)]
= a2b2 + a2c2 + a2c2 + b2c2 + b2c2 + d 2a2 + d 2b2 + d 2b2 + d 2c2
= a2b2 + a2c2 + a2d 2 + b2 x b2 + b2c2 + b2d 2 + c2b2 + c2 x c2 + c2d 2
Using (2) and (3) and rearranging the terms
RHS = a2 (b2 + c2 + d 2) + b2 (b2 + c2 + d 2) + c2 (b2 + c2 + d 2)
= (a2 + b2 + c2)(b2 + c2 + d 2)
= LHS
Thus, (a2 + b2 + c2)(b2 + c2 + d 2) = (ab + bc + cd)2 Proved
NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.3 Question 25
If a, b, c an d are in G.P. Show that: (a2 + b2 + c2)(b2 + c2 + d 2) = (ab + bc + cd)2
Summary:
Given that a, b, c, d in G.P we showed that a2 + b2 + c2)(b2 + c2 + d 2) = (ab + bc + cd)2
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