Understand the cycle of powers of $i$

The powers of $i$ repeat every $4$:

and then the pattern repeats.

So the value of $i^k$ depends only on $k \pmod 4$.

When is $i^{k_1}+i^{k_2}=0$?

For the sum to be zero, we need the two terms to be additive inverses:

From the cycle $\{1,i,-1,-i\}$, each value is negated by shifting exponent by $2$ modulo $4$:

Hence,

(equivalently, $k_1$ and $k_2$ differ by $2$ modulo $4$).

Find the probability of this happening

A randomly chosen natural number is equally likely to be in any residue class modulo $4$:

So for each of $k_1$ and $k_2$, there are $4$ equally likely residue classes.

Total possible residue pairs:

Favorable pairs for zero sum satisfy difference $2$ modulo $4$:

So there are $4$ such pairs.

Therefore,

Required probability: non-zero sum

Check options

• A: $\frac34$ ✅
• B: $\frac12$
• C: $\frac14$
• D: $\frac23$

Therefore, the correct option is A.