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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:

$T_{r+1}={ }^{4 n-2} C_r(i x)^r$
$T_{r+1}$ is negative, if $i^r$ is negative and real.
$i^r=-1$
$\Rightarrow r=2,6,10, \ldots$ which form an A.P.
$0 \leq r \leq 4 n-2$
$4 n-2=2+(r-1) 4$
$\Rightarrow r=n$
The required number of terms is $n$.