Question

If i=√-1 and x is a positive integer, then i^n+i^n+1+i^n+2+i^n+3 is equal to​

Answer 1

Answer:

0

Step-by-step explanation:

Given that $i=\sqrt{-1}$

On squaring it, we get $i^2=-1$

On cubing,

$i^3=i^{2+1}=i^2\times i=-1\times i=-i$

On taking 4th power,

$i^4=(i^2)^2=(-1)^2=1$

Now come to the question.

[Note: x is given in the question mistakenly, instead n is there.]

$i^n+i^{n+1}+i^{n+2}+i^{n+3}\\ \\ \Longrightarrow\ i^n+i^n\times i+i^n\times i^2+i^n\times i^3$

We take i^n common from each.

$i^n(1+i+i^2+i^3)$

Now,

$i^n(1+i+i^2+i^3)\\ \\ \Longrightarrow\ i^n(1+i-1-i)\ \ \ \ \ \ \ \ \ \ [\because\ i^2=-1\ \ \&\ \ i^3=-i]\\ \\ \Longrightarrow\ i^n\times 0\\ \\ \Longrightarrow\ \mathbf{0}$

Hence 0 is the answer.

And also it is remembered that "the sum of four consecutive integral powers of i is always 0."