= Solution
Write $S^n$ as the union of slightly enlarged northern and southern hemispheres $U$ and $V$. Both are <contractible space>[contractible], while $U\cap V$ deformation retracts onto $S^{n-1}$. The reduced <Mayer-Vietoris theorem> therefore gives
$$
\widetilde H_i(S^n;\mathbb Z)\cong\widetilde H_{i-1}(S^{n-1};\mathbb Z).
$$
Starting from $\widetilde H_0(S^0;\mathbb Z)\cong\mathbb Z$ proves the <homology of a sphere>:
$$
H_i(S^n;\mathbb Z)\cong
\begin{cases}
\mathbb Z,&i=0,n,\\
0,&\text{otherwise}.
\end{cases}
$$
This uses no <cellular homology>. A reflection of $S^n$ reverses its orientation and has <degree of a continuous mapping> $-1$, so it induces the identity on $H_0$, multiplication by $-1$ on $H_n$, and the unique map between zero groups in every other degree.
For a <CW complex> with skeleta $X^k$, its <cellular chain complex> is
$$
C_k^{\mathrm{cell}}(X)=H_k(X^k,X^{k-1};\mathbb Z)\cong\bigoplus_{\text{$k$-cells}}\mathbb Z.
$$
The differential is the connecting map to $H_{k-1}(X^{k-1})$ followed by passage to $H_{k-1}(X^{k-1},X^{k-2})$. Equivalently, the coefficient of a $(k-1)$-cell in the boundary of a $k$-cell is the <degree of a continuous mapping>[degree] obtained from its attaching map after collapsing the complement of that lower cell. This is the <cellular boundary formula>.
The quotient $D^k/(x\sim-x\text{ on }S^{k-1})$ builds $\mathbb{RP}^k$ from $\mathbb{RP}^{k-1}$ by one $k$-cell, so $\mathbb{RP}^n$ has one cell in each dimension $0,\ldots,n$. The two lifts of the attaching map contribute with relative sign $(-1)^k$, and the <cellular homology of real projective space> has differential
$$
d_k=1+(-1)^k=
\begin{cases}
0,&k\text{ odd},\\
2,&k\text{ even}.
\end{cases}
$$
Consequently
$$
H_i(\mathbb{RP}^n;\mathbb Z)\cong
\begin{cases}
\mathbb Z,&i=0,\\
\mathbb Z/2,&0<i<n\text{ and }i\text{ odd},\\
\mathbb Z,&i=n\text{ and }n\text{ odd},\\
0,&\text{otherwise}.
\end{cases}
$$
With $\mathbb F_2=\mathbb Z/2$ coefficients every differential vanishes, and hence
$$
\boxed{H_i(\mathbb{RP}^n;\mathbb F_2)\cong\mathbb F_2\quad(0\leq i\leq n),}
$$
with zero homology outside that range.
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