Step 1: Understanding the Concept:
To find the magnitude (modulus) of the sum, we must first simplify \( z_1 \) by evaluating the powers of \( i \). Powers of \( i \) repeat in a cycle of four: \( i^1=i, i^2=-1, i^3=-i, i^4=1 \).
Step 2: Key Formula or Approach:
1. \( i^{4n} = 1, i^{4n+1} = i, i^{4n+2} = -1, i^{4n+3} = -i \).
2. \( |a + bi| = \sqrt{a^2 + b^2} \).
Step 3: Detailed Explanation:
Simplify \( z_1 \):
\[ i^{40} = (i^4)^{10} = 1^{10} = 1 \]
\[ i^{35} = i^{32} \cdot i^3 = 1 \cdot (-i) = -i \]
\[ i^{17} = i^{16} \cdot i^1 = 1 \cdot i = i \]
Substituting these into \( z_1 \):
\[ z_1 = 4(1) - 5(-i) + 6(i) + 2 = 4 + 5i + 6i + 2 = 6 + 11i \]
Now find \( z_1 + z_2 \):
\[ z_1 + z_2 = (6 + 11i) + (-1 + i) = 5 + 12i \]
Calculate the modulus:
\[ |z_1 + z_2| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
Step 4: Final Answer:
The modulus \( |z_1 + z_2| \) is 13.