Symetric powers of $sl_2$ representations

Question

I'd like to understand some special things about representations of $sl_2$ (which is considered as a Lie algebra over $\mathbb{C}$). First, it can be shown that for each $n\in \mathbb{N}$ there is only one $n$-dimensional irreducible representation of $sl_2$: let me denote it by $V_n$. Could you please help me to understand the following identities? $$\mathrm{Sym}^2 V_n=\bigoplus_{i=0}^{[n/2]} V_{2n-4i}$$ $$\mathrm{Sym}^k V_n\simeq \mathrm{Sym}^n V_k$$ I will be very grateful for either a hint or a link for a proof!