The Betti number of complex projective spaces

Question

I have known from the wikipedia that the Betti number of the complex projective space run 1, 0, 1, 0, ..., 0, 1, 0, 0, 0, ...

That is, 0 in odd dimensions, 1 in even dimensions up to 2n. 
However,I don't know why it is that and how to compute the homology groups for $Cp^n$

Answer 1

Hint : if you know cellular homology, decompose $\Bbb CP^n$ into even-dimensional cells. The complex looks like $C_{2n} \to 0 \to \dots C_2 \to 0 \to C_0$ so it follows that the homology is freely generated in even degree, the Betti number $b_k$ being the number of $2k$ cells.