Question

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

Answer 1

Hint: 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 letter$i$. 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)${i^{4n}} = 1$
Let $n = 1$
${i^{4n}} = {i^{4 \times 1}}$
$ = {i^4}$
$ = 1$
Hence it is the correct option.
2)${i^{4n - 1}} = i$
$ = {i^{4 \times 1 - 1}}$
$ = {i^3}$
$ = {i^2} \times i$
$ = - 1 \times j$
$ = - i$
But it is given that${i^{4n - 1}} = i$.
Since we obtain${i^{4n - 1}} = - i$,${i^{4n - 1}} \ne i$
Hence the given option is false.
3)${i^{4n + 1}} = i$
Let $n = 1$
${i^{4n + 1}} = {i^{4 \times 1 + 1}}$
$ = {i^{4 + 1}}$
$ = {i^5}$
$ = {i^4} \times i$
$ = 1 \times i$
$ = i$
Hence it is the correct option.
4)${i^{4n}} = 1$
Let $n = 1$
${i^{4 \times 1}} = {i^4}$
$ = 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 letter$i$. Here, we need to analyze every option so that we are able to identify the false option. When we substitute$n = 1$in every option, we are able to choose the wrong option that is the required answer.