NCERT Class XI Mathematics - Sequences and Series - Solutions
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Question : 59
Total: 106
If a, b, c and d are in G.P., show that
( a 2 + b 2 + c 2 ) ( b 2 + c 2 + d 2 ) ( a b + b c + c d ) 2
Solution:
We have a, b, c, d are in G.P.
Let r be a common ratio, then b = ar, c =a r 2 a r 3
∴( a 2 + b 2 + c 2 ) ( b 2 + c 2 + d 2 )
=[ a 2 + ( a r ) 2 + ( a r 2 ) 2 ] [ ( a r ) 2 + ( a r 2 ) 2 + ( a r 3 ) 2 ] [ a 2 + a 2 r 2 + a 2 r 4 ] [ a 2 r 2 + a 2 r 4 + a 2 r 6 ]
=a 2 ( 1 + r 2 + r 4 ) a 2 r 2 ( 1 + r 2 + r 4 )
=a 4 r 2 ( 1 + r 2 + r 4 ) 2
Also,( a b + b c + c d ) 2 ( a 2 r + a 2 r 3 + a 2 r 5 ) 2
=a 4 r 2 ( 1 + r 2 + r 4 ) 2
From (i) & (ii), we get( a 2 + b 2 + c 2 ) ( b 2 + c 2 + d 2 ) ( a b + b c + c d ) 2
Let r be a common ratio, then b = ar, c =
∴
=
=
=
Also,
=
From (i) & (ii), we get
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