The number of real negative terms in the binomial expansion of (1+i x)4 n-2, n ∈ N , x0 is

Question

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

Tardigrade Admin · Accepted Answer

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 ≤ r ≤ 4 n -2 and 4 n-2=2+(p-1) 4 [pth term of A.P.] ⇒ p=n Hence, required number of terms = n

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

n

n + 1

n - 1

2 n

Here, $T_{r + 1} =^{4 n - 2} C_{r} (i x)^{r}$
The term is real negative if $r = 2, 6, 10, \dots\dots$
but $0 \leq r \leq 4 n - 2$
and $4 n - 2 = 2 + (p - 1) 4$ [pth term of A.P.] $\Rightarrow p = n$
Hence, required number of terms $= n$

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

n

n + 1

n - 1

2 n

Here, Tr+1​=4n−2Cr​(ix)r The term is real negative if r=2,6,10,…… but 0≤r≤4n−2 and 4n−2=2+(p−1)4 [pth term of A.P.] ⇒p=n Hence, required number of terms =n

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

Here, $T_{r + 1} =^{4 n - 2} C_{r} (i x)^{r}$
The term is real negative if $r = 2, 6, 10, \dots\dots$
but $0 \leq r \leq 4 n - 2$
and $4 n - 2 = 2 + (p - 1) 4$ [pth term of A.P.] $\Rightarrow p = n$
Hence, required number of terms $= n$