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Question

If a1,a2,...an are in H.P. then the expression a1a2+a2a3+...+an1anisequalto

A
(n1)a1an
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B
(n1)(a1an)
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C
n(a1an)
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D
na1an
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Solution

The correct option is C (n1)a1an
a1,a2,a3an are in HP
1a1,1a2,1a31an are in AP
1a21a1=da1a2a1a2=d
a1a2=a1a2d
a1a2=a1a2d
Similarly ,a2a3=a2a3dan1an=an1and
a1a2+a2a3+an1an=a1a2d+a2a3d++an1and
=1d(a1α2+a2+an1an)
=a1and
To find "d
1an=1a1+(n1)d
1an=1a1+ndd
1an1a1=d(n1)
a1an(a1an)(n1)=d
a1a2+a2a3+an1an=a1and
=a1ana1an(a1an)(n1)
=(n1)(a1αn)(a1an)(a1gn)
=(n1)(a1an)
Answer: option (A)

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