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Question

If a1,a2,a3,......an are in H.P then a1a2+a2a3+a3a4....+an1an= ?

A
(n1)a1an
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B
(n+1)a1an
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C
(n2)a1an
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D
(n+4)a1an
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Solution

The correct option is C (n1)a1an
Since a1,a2,a3,...an are in H.P
1a1,1a2,1a3,...,1an are in A.P
Let d be the common difference of AP
1a21a1=d
a1a2a1a2=d
a1a2=a1a2d
a2a3=a2a3d
..............................................
an1an=an1and
On adding all of these , we get
a1an=d(a1a2+a2a3+...+an1an) .......(1)
Also,1an=1a1+(n1)d
d=a1ana1an(n1)
On putting the value of d in Eq, (i), we get
a1an=a1ana1an(n1)(a1a2+a2a3+...+an1an)
a1a2+a2a3+...+an1an=a1an(n1)


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