Question

If we have ${{\left( -1 \right)}^{n}}+{{\left( -1 \right)}^{4n}}=0$ , then n is:
(A). Any positive integer
(B). Any negative integer
(C). Any odd natural number
(D). Any even natural number.

Answer 1

Hint: For solving the above question you need to focus on the point that -1 to the power any even number is positive and equal to 1 and -1 to the power any odd number is equal to -1 which is negative.

Complete step-by-step solution -
Let us start the solution to the above question, by rearranging the terms according to our need.
${{\left( -1 \right)}^{n}}+{{\left( -1 \right)}^{4n}}=0$
We will take ${{\left( -1 \right)}^{4n}}$ to the other side of the equation. On doing so, we get
${{\left( -1 \right)}^{n}}=-{{\left( -1 \right)}^{4n}}$
Now we know that 4n is an even number, as 4n is divisible by 4 which implies that it is also divisible by 2. So, ${{\left( -1 \right)}^{4n}}$ is always positive, irrespective of the value of n. Therefore, we can say that the right-hand side of the equation is always negative and is equal to -1.
Now for the equation to be true, the right-hand side must be equal to the left-hand side of the equation. So, ${{\left( -1 \right)}^{n}}$ must be negative. For this to be negative, n must be an odd number, as a negative number to the power of an odd integer is negative. So, we can say that n is an odd natural number.
Hence, the answer to the above question is option (c).

Note: The answer to the above question is the integers whose absolute value is odd, but as the options only talks about the natural number, so we go with the most suitable option. However, if we take n to be -3 , -1, -5………… , they would also satisfy the relation.