if n is any integer ,then in is (a) i,(b) 1,-1 (c) i,-i (d) 1,-1,i,-i
If you mean i^n, then (d)
yes
To determine the possible values of in, where n is any integer, we can analyze the pattern of powers of i.
The imaginary unit i is defined as the square root of -1. When i is raised to powers, a periodic pattern emerges:
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
As you can see, after i^4, the pattern repeats itself. Therefore, any positive integer power of i that is a multiple of 4 will yield the value 1. The other values -1, i, and -i can be obtained by raising i to the first three positive integers modulo 4.
In this case, since n can be any integer, we need to consider all possibilities.
(a) When n is a multiple of 4, the power of i will be 1. For example, i^4, i^8, i^12, etc. These powers will always yield 1.
(b) When n is 2 more than a multiple of 4, the power of i will be -1. For example, i^2, i^6, i^10, etc. These powers will always yield -1.
(c) When n is 3 more than a multiple of 4, the power of i will be -i. For example, i^3, i^7, i^11, etc. These powers will always yield -i.
(d) When n is 1 more than a multiple of 4, the power of i will be i. For example, i^1, i^5, i^9, etc. These powers will always yield i.
Therefore, the possible values of in, where n is any integer, are (a) 1, (b) -1, (c) -i, (d) i.