We work over the field of complex numbers. Consider the weighted projective space $\mathbb P(1^n,2)$, that is given by $$ \frac{\mathbb A^{n+1} \setminus \{0\}}{\mathbb C^*} $$ where $\mathbb C^*$ acts on the affine space $\mathbb A^{n+1}$ as $$ t \cdot (x_0,\ldots,x_n) =(tx_0,\ldots,tx_{n-1},t^2x_n). $$ It can be proved (see the notes of Miles Reid about weighted projective spaces) that $\mathbb P(1^n,2)$ is isomorphic to a cone over $\nu_2\mathbb P^{n-1}$, where $\nu_2:\mathbb P^{n-1} \hookrightarrow \mathbb P(H^0(\mathbb P^{n-1},\mathcal O(2))^\vee)$ is the second Veronese embedding.
It has been told me that the normal bundle of $\nu_2 \mathbb P^{n-1}$ inside the cone $\mathbb P(1^n,2)$ is the line bundle $\mathcal O(2)$, but I cannot find a proof of this fact. Any idea?