Let $f : S^{2n+1} \rightarrow \mathbb C P^n$ be a continuous map and $X := \mathrm{Cone}(S^{2n+1}) \cup_f \mathbb C P^n$.
We have the following long exact sequence:
$$\cdots \rightarrow \pi_{2n+2}(X) \rightarrow \pi_{2n+2}(X, \mathbb C P^n) \rightarrow \pi_{2n+1}(\mathbb C P^n) \rightarrow \pi_{2n+1}(X) \rightarrow \pi_{2n+1}(X, \mathbb C P^n) \rightarrow \cdots$$
My guess is that $\pi_{2n+2}(X, \mathbb C P^n) \cong \pi_{2n+1}(S^{2n+1}) \cong \mathbb Z$ and $\pi_{2n+2}(X) = \pi_{2n+1}(X) = 0$, which would imply $\pi_{2n+1}(\mathbb C P^n) \cong \mathbb Z$.
Is my guess correct? If so, could you help me with proving these?