Certainly! Let's delve into the problem step by step and analyze what happens to [tex]\( f(n) \)[/tex] as [tex]\( n \)[/tex] increases.
### Step 1: Understand the function
We are given:
[tex]\[ f(n) = \left|\left(\frac{5}{8} + \frac{3}{4}i\right)^n\right| \][/tex]
This is the magnitude of a complex number raised to the power [tex]\( n \)[/tex].
### Step 2: Calculate the Magnitude of the Complex Number
To understand the behavior of [tex]\( f(n) \)[/tex], we need to start by determining the magnitude of the base complex number:
[tex]\[ z = \frac{5}{8} + \frac{3}{4}i \][/tex]
The magnitude (or modulus) of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[ |a + bi| = \sqrt{a^2 + b^2} \][/tex]
Here, [tex]\( a = \frac{5}{8} \)[/tex] and [tex]\( b = \frac{3}{4} \)[/tex]. Let's find the magnitude:
[tex]\[
|z| = \left| \frac{5}{8} + \frac{3}{4} i \right| = \sqrt{\left(\frac{5}{8}\right)^2 + \left(\frac{3}{4}\right)^2}
\][/tex]
### Step 3: Perform the Calculation
Calculate each term separately:
[tex]\[
\left(\frac{5}{8}\right)^2 = \frac{25}{64}
\][/tex]
[tex]\[
\left(\frac{3}{4}\right)^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16} = \frac{36}{64}
\][/tex]
Now, add the two results:
[tex]\[
\frac{25}{64} + \frac{36}{64} = \frac{61}{64}
\][/tex]
Taking the square root:
[tex]\[
|z| = \sqrt{\frac{61}{64}} = \frac{\sqrt{61}}{8}
\][/tex]
### Step 4: Analyze the Magnitude
Now, let’s examine the magnitude to understand the behavior of [tex]\( f(n) \)[/tex]:
- Given [tex]\( \frac{\sqrt{61}}{8} \)[/tex]
We know that:
[tex]\[ \sqrt{61} \approx 7.81 \][/tex]
Thus:
[tex]\[ \frac{\sqrt{61}}{8} \approx \frac{7.81}{8} \approx 0.97625 \][/tex]
### Step 5: Determine the Behavior of [tex]\( f(n) \)[/tex]
We can see that:
[tex]\[ |a + bi| = \frac{\sqrt{61}}{8} < 1 \][/tex]
Since the magnitude of the complex number is less than 1, raising it to higher powers will result in a decrease in the magnitude:
[tex]\[ \left(\frac{\sqrt{61}}{8}\right)^n \][/tex]
### Conclusion
Therefore, as [tex]\( n \)[/tex] increases, [tex]\( f(n) = \left|\left(\frac{5}{8} + \frac{3}{4}i\right)^n\right| \)[/tex] will decrease.
The correct answer is:
[tex]\[ \boxed{2} \][/tex]
Hence, the behavior of [tex]\( f(n) \)[/tex] as [tex]\( n \)[/tex] increases is described by:
[tex]\[ \text{(B) As } n \text{ increases, } f(n) \text{ decreases.} \][/tex]
