option C
$P_n$ is event that odd number of heads after occur $n$ tosses.
$P_0=0$
$P_1=p (H)$
$P_2=2p(1-p) (HT, TH)$
$P_3=3p(1-p)(1-p)+p^3 (HTT,HHH,THT,TTH)$
$p_4=4(p^{3})(1-p)+4p(1-p)^{3}$ (HHHT(4 permutations), HTTT(4 permutations))
Clearly option A,E is false, option B is false for $P_1,P_2$...(check by substituting values)
Take $P_2$, using given equation in option B we get $P_2=2p^2-3p+2$ but actual answer is $2p(1-p)$
option D is false as $P_0=0$.