Question

If $n₁, n₂$ are positive integers, then $(1+i)ⁿ₁+(1+i³)ⁿ₁+(1+i⁵)ⁿ₂+(1+i⁷)ⁿ₂$ is a real number if and only if

• A. $n_1 = n_2 + 1$
• B. $n_1 = n_2$
• C. $n_1, n_2$ are any two negative integers
• D. $n_1, n_2$ are both any positive integers

Hint:

If $n1, n2$ are positive integers, then $(1+i)+(1+i)+(1+i)$ is a real number if and only if

Answer 1

The Correct Option is D

Step 1: Concept
Simplify powers of $i$: $i^3 = -i$, $i^5 = i$, $i^7 = -i$.
Step 2: Analysis
The expression is $(1+i)^{n_1} + (1-i)^{n_1} + (1+i)^{n_2} + (1-i)^{n_2}$. Note that $(1-i)$ is the conjugate of $(1+i)$.
Step 3: Evaluation
Using polar form $z + \bar{z} = 2 \text{Re}(z)$, the sum $(1+i)^n + (1-i)^n = 2^{n/2} \cdot 2\cos(\frac{n\pi}{4})$, which is always real for any integer $n$.
Step 4: Conclusion
Since both parts of the sum are real for any positive integers $n_1$ and $n_2$, the entire expression is real.
Final Answer: (d)