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Question

If a1,a2,a3,...an are in H.P then
a1a2+a3+...an,a2a1+a3+...an,a3a1+a2+a4+...+an,ana1+a2+...+an are in

A
A.P
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B
GP
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C
H.P
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D
A.G.P
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Solution

The correct option is C H.P
If a1,a2,a3,...an are in H.P then
1a1,1a2,1a3,...1an in A.P
Also,
a1+a2+a3+...ana1,a1+a2+a3+...ana2,a1+a2+a3+...ana3,...a1+a2+a3+...anan in A.P
1+a2+a3+...ana1,1+a1+a3+...ana2,1+a1+a2+...ana3,...1+a1+a2+a3,...an in A.P
a2+a3+...ana1,a1+a3+...ana2,a1+a2+...ana3,...a1+a2+a3,...an in A.P
Therefore, reciprocal of it will be in H.P
a1a2+a3+...an,a2a1+a3+...an,a3a1+a2+a4+...+an,ana1+a2+...+an

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