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  4. The number of real negative terms in the binomial expansion of (1+i x)4 n-2, n ∈ N , x>0 is

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Q. The number of real negative terms in the binomial expansion of $(1+i x)^{4 n-2}, n \in N , x>0$ is

Binomial Theorem

Solution:

Here, $T _{ r +1}={ }^{4 n -2} C _{ r }( ix )^{ r }$
The term is real negative if $r =2,6,10, \ldots \ldots$
but $ 0 \leq r \leq 4 n -2$
and $4 n-2=2+(p-1) 4 \quad$ [pth term of A.P.] $\Rightarrow p=n$
Hence, required number of terms $= n$