14. If (a + b)2 + (b + c)2 + (c + d...
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14. If (a + b)2 + (b + c)2 + (c + d)2 = 4(ab + bc + cd) prove that a = b = c = d.

JEE/Engineering Exams
Maths
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( (a+b)^{2}+(b+c)^{2}+(c+d)^{2}=4(a b+b c+c d) quad(g sin e n] ) ( a^{2}+b^{2}+2 a b+b^{2}+c^{2}+2 b c+c^{2}+a^{2}+2 c d=4(a b+b c+c a) ) ( a^{2}+2 b^{2}+2 c^{2}+d^{2}+2(a b+b c+c d)=4(a b+b c+c d) ) ( a^{2}+2left(b^{2}+c^{2}right)+d^{2}=(4 a b+4 b c+4 c d)-2(a b+b c+c d ) ( a^{2}+b^{2}+b^{2}+c^{2}+c^{2}+d^{2}=2 a b+2 b c+2 c d ) ( left(a^{2}+b^{2}-2 a bright)+left(b^{2}+c^{2}-2 b cright)+left(c^{2}+d^{2}-2 c dright)=0 ) ( (a-b)^{2}+(b-c)^{2}+(c-d)^{2}=0 ) ( Rightarrow quad(a-b)^{2}=0 quad Rightarrow quad a=b quad-0 ) ( Rightarrow quad(b-e)^{2}=0 quad Rightarrow quad b>c quad rightarrow(2) ) ( Rightarrow quad(c-d)^{2}=0 quad Rightarrow quad c=d ) from ( 0,0 times 0 ) ( a=b=c= )
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