Question

If $z_1 = 4i^{40} - 5i^{35} + 6i^{17} + 2$ and $z_2 = -1 + i$, then $|z_1 + z_2| =$

• A. 13
• B. 5
• C. 15
• D. 12

Hint:

$i^n$ repeats every 4 powers. Divide the exponent by 4 and use the remainder.

Answer 1

The Correct Option is A

Step 1: Concept
Simplify powers of $i$ using $i^4 = 1, i^3 = -i, i^2 = -1, i^1 = i$.

Step 2: Meaning
$z_1 = 4(1) - 5(-i) + 6(i) + 2 = 4 + 5i + 6i + 2 = 6 + 11i$.

Step 3: Analysis
$z_1 + z_2 = (6 + 11i) + (-1 + i) = 5 + 12i$.

Step 4: Conclusion
$|z_1 + z_2| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
Final Answer: (A)