For positive integers n_1, n_2, the value of | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the expression: $(1 + i)^{n_1} + (1 + i^3)^{n_1} + (1 + i^5)^{n_2} + (1 + i^7)^{n_2}$.Since $i^3 = -i$, $i^5 = i$, and $i^7 = -i$, the expre

Vedclass · Accepted Answer

$n_1 > 0, n_2 > 0$

Vedclass · Answer

(D) Given the expression: $(1 + i)^{n_1} + (1 + i^3)^{n_1} + (1 + i^5)^{n_2} + (1 + i^7)^{n_2}$.Since $i^3 = -i$, $i^5 = i$, and $i^7 = -i$, the expression becomes:$(1 + i)^{n_1} + (1 - i)^{n_1} + (1 + i)^{n_2} + (1 - i)^{n_2}$.Let $z = (1 + i)^{n} + (1 - i)^{n}$.Using the polar form $1 + i = \sqrt{2} e^{i\pi/4}$ and $1 - i = \sqrt{2} e^{-i\pi/4}$, we get:$z = (\sqrt{2})^n (e^{in\pi/4} + e^{-in\pi/4}) = 2^{n/2} \cdot 2 \cos(n\pi/4) = 2^{n/2+1} \cos(n\pi/4)$.Since $2^{n/2+1} \cos(n\pi/4)$ is always a real number for any positive integer $n$, the sum of two such terms is also always real.Therefore, the expression is a real number for all positive integers $n_1$ and $n_2$.

For positive integers $n_1, n_2$, the value of the expression $(1 + i)^{n_1} + (1 + i^3)^{n_1} + (1 + i^5)^{n_2} + (1 + i^7)^{n_2}$, where $i = \sqrt{-1}$, is a real number if and only if:

Similar Questions

Let $a, b$ be two real numbers such that $ab < 0$. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$,then a possible value of $\frac{1+[a]}{4b}$,where $[t]$ is the greatest integer function,is:

If $x = 1 + 2i$,then the value of $x^3 + 7x^2 - x + 16$ is

The imaginary part of $\cosh(\alpha + i\beta)$ is

Let a complex number $z$,$|z| \neq 1$,satisfy $\log_{\frac{1}{\sqrt{2}}} \left( \frac{|z|+11}{(|z|-1)^2} \right) \leq 2$. Then,the largest value of $|z|$ is equal to ............

If $z=3+5i$,then $z^3+\bar{z}+198$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module