The correct option is
C (n−1)a1anSince a1,a2,a3,...an are in H.P
∴1a1,1a2,1a3,...,1an are in A.P
Let d be the common difference of AP
∴1a2−1a1=d
⇒a1−a2a1a2=d
⇒a1−a2=a1a2d
⇒a2−a3=a2a3d
..............................................
an−1−an=an−1and
On adding all of these , we get
a1−an=d(a1a2+a2a3+...+an−1an) .......(1)
Also,1an=1a1+(n−1)d
⇒d=a1−ana1an(n−1)
On putting the value of d in Eq, (i), we get
a1−an=a1−ana1an(n−1)(a1a2+a2a3+...+an−1an)
⇒a1a2+a2a3+...+an−1an=a1an(n−1)