A particle executing `SHM` from exterme position towards centre is observed to be at distances `x_(1),x_(2)` and `x_(3)` from the centre at the end of three successive seconds. The period of `SHM` is.

Explanation

`x_(1) = A cos omega, x_(2) = A cos 2 omega ,x_(3) =A cos 3 omega`
`x_(1) +x_(3) =A [cos omega + cos 3 omega]`
from, `cos A +cos B = 2 cos ((A+B)/(2)) cos((A-B)/(2))`
`x_(1) +x_(3) = A[2 cos omega cos 2 omega]`
`(x_(1)+x_(3))/(2x_(2)) = cos omega rArr omega = cos^(-1) [(x_(1)+x_(3))/(2x_(2))]`
`T = (2pi)/(cos^(-1)[(x_(1)+x_(2))/(2x_(2))])`