Final answer:
The function f(n) = |(1/2 + 7/8 i)^n| increases as n increases, because the magnitude of the complex number (1/2 + 7/8 i) is greater than 1. The correct answer is option c.
Step-by-step explanation:
The question asks what happens to the value of the function f(n) = |(1/2 + 7/8 i)^n| as n increases.
To determine the behavior of f(n), it is essential to understand the properties of complex numbers and their magnitudes.
The magnitude of a complex number a + bi is given by
|a + bi| = √(a^2 + b^2).
In this case, the magnitude of 1/2 + 7/8 i can be calculated as follows:
|1/2 + 7/8 i| = √((1/2)^2 + (7/8)^2) = √((1/4) + (49/64)) = √(√((64/64)+( a = √(√((113/64))) = √(1.765625)) = 1.33 (approximately)
The magnitude of this complex number is greater than 1, so each time we exponentiate it (increase n), the overall magnitude will increase.
Therefore, the absolute value of (1/2 + 7/8 i)^n will also increase as n increases, meaning f(n) increases as n increases. The correct answer to the original question is 'c. As n increases, f(n) increases.'