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Question

If a1,a2,a3 are in A.P., a2,a3,a4 are in G.P., and a3,a4,a5 are in H.P., then lna1,lna3,lna5 are in


A
G.P.
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B
A.P.
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C
H.P.
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D
None of these
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Solution

The correct option is A A.P.
Given:
a1,a2,a3 are in A.P.
2a2=a1+a3(1)
a2,a3,a4 are in G.P.
a23=a2a4(2)
a3, a4, a5 are in H.P.
a4=2a3a5a3+a5(3)
Using equation (2) and (3),
a23a2=2a3a5a3+a5a3(a3+a5)=2a2a5
Using equation (1),
a23+a3a5=(a1+a3)a5a23=a1a5
Taking ln on both sides,
2lna3=ln(a1a5)2lna3=lna1+lna5

Hence, lna1,lna3,lna5 are in A.P.

Mathematics

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