Given the sequence of numbers `x_(1),x_(2),x_(3),x_(4),….,x_(2005)`, `(x_(1))/(x_(1)+1)=(x_(2))/(x_(2)+3)=(x_(3))/(x_(3)+5)=...=(x_(2005))/(x_(2005)+4009)`, the nature of the sequence is

Options

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

Answer

A

Explanation

`(a)` Given `(x_(1))/(x_(1)+1)=(x_(2))/(x_(2)+3)=(x_(3))/(x_(3)+5)=...=(x_(2005))/(x_(2005)+4009)`
`impliesx_(1)=(lambda)/(1-lambda)`, `x_(2)=(3lambda)/(1-lambda)`, `x_(3)=(5lambda)/(1-lambda)`,……
Hence , `x_(1),x_(2),x_(3),…..,x_(2005)` are in arithmetic progression.