Final answer:
As n increases, the function f(n)=|(0.5 + 0.2i)^n| approaches zero because the modulus of the complex number (0.5 + 0.2i) is less than 1, meaning each subsequent multiplication makes the product smaller.
Step-by-step explanation:
The student is asking about the behavior of the function f(n)=|(0.5 + 0.2i)^n| as n increases. This function involves a complex number raised to the power of n and then taking the absolute value (also known as the modulus) of the result. As n increases, the complex number (0.5 + 0.2i) will be raised to higher and higher powers. The modulus of a complex number (a + bi) is √(a^2 + b^2), so for our given complex number the initial modulus is √(0.5^2 + 0.2^2).
Since the modulus of (0.5 + 0.2i) is less than 1, repeatedly multiplying it by itself will result in a value that approaches zero. Therefore, as n approaches infinity, the function f(n) approaches zero. This is consistent with the behavior of geometric sequences with a common ratio whose absolute value is less than 1.