A particle executing $SHM$ while moving from the one extremity is found at distance $x_1, x_2$ and $x_3$ from the centre at the end of three successive seconds. The time period of oscillation is (θ) used in the following choices is given by cosθ=x1+x3/2x2

The correct option is $\mathbf{A}$

$\frac{2 \pi}{\theta}$

$\begin{array}{l} \mathrm{x}_{1}=A \cos (\omega) \\ \mathrm{x}_{2}=A \cos (2 \omega) \\ \mathrm{x}_{3}=A \cos (3 \omega) \\ \therefore \mathrm{x}_{1}+\mathrm{x}_{3}=2 A \cos (2 \omega) \cos \omega=2 A \frac{x_{2} x_{1}}{A A} \\ \Rightarrow \frac{x_{1}}{A}=\frac{x_{1}+x_{2} x_{1}}{2 x_{2} A} \end{array}$

Hence $T=\frac{2 \pi}{\omega}=\frac{2 \pi}{\theta}$