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Question: If \(n\) is a positive integer, then which of the following relations is false. 1\. \({i^{4n}} = 1...

If nn is a positive integer, then which of the following relations is false.
1. i4n=1{i^{4n}} = 1
2. i4n−1=i{i^{4n - 1}} = i
3. i4n+1=i{i^{4n + 1}} = i
4. i4n=1{i^{4n}} = 1

Explanation

Solution

First, we shall analyze the given information so that we can able to solve the given problem. Here, we are asked to choose the wrong option which contains some relations. While squaring a number results in a negative result, it can be known as imaginary numbers and it is denoted by a letterii. A complex number is a number that is formed due to the combination of a real number and an imaginary number.

Complete step by step answer:
1)i4n=1{i^{4n}} = 1
Let n=1n = 1
i4n=i4×1{i^{4n}} = {i^{4 \times 1}}
=i4= {i^4}
=1= 1
Hence it is the correct option.
2)i4n−1=i{i^{4n - 1}} = i
=i4×1−1= {i^{4 \times 1 - 1}}
=i3= {i^3}
=i2×i= {i^2} \times i
=−1×j= - 1 \times j
=−i= - i
But it is given thati4n−1=i{i^{4n - 1}} = i.
Since we obtaini4n−1=−i{i^{4n - 1}} = - i,i4n−1≠i{i^{4n - 1}} \ne i
Hence the given option is false.
3)i4n+1=i{i^{4n + 1}} = i
Let n=1n = 1
i4n+1=i4×1+1{i^{4n + 1}} = {i^{4 \times 1 + 1}}
=i4+1= {i^{4 + 1}}
=i5= {i^5}
=i4×i= {i^4} \times i
=1×i= 1 \times i
=i= i
Hence it is the correct option.
4)i4n=1{i^{4n}} = 1
Let n=1n = 1
i4×1=i4{i^{4 \times 1}} = {i^4}
=1= 1
Hence it is the correct option.

So, the correct answer is “Option 2”.

Note: While squaring a number results in a negative result, it can be known as imaginary numbers and it is denoted by a letterii. Here, we need to analyze every option so that we are able to identify the false option. When we substituten=1n = 1in every option, we are able to choose the wrong option that is the required answer.