If `I_(n)=int_(0)^(pi//4) tan^(n)x dx`, then
`(1)/(I_(2)+I_(4)),(1)/(I_(3)+I_(5)),(1)/(I_(4)+I_(6)),...` from\

Options

A. am A.P.
B. a G.P.
C. a H.P.
D. none of these

Explanation

To solve the problem, we need to evaluate the integral \( I_n = \int_0^{\frac{\pi}{4}} \tan^n x \, dx \) and derive the relationships between \( I_n \) for different values of \( n \).

### Step-by-Step Solution:

1. **Understanding the Integral**:
We start with the integral \( I_n = \int_0^{\frac{\pi}{4}} \tan^n x \, dx \). We will derive a recurrence relation for \( I_n \).

2. **Using Integration by Parts**:
We can express \( I_n \) in terms of \( I_{n-2} \) using the identity:
\[ I_n = \int_0^{\frac{\pi}{4}} \tan^n x \, dx = \int_0^{\frac{\pi}{4}} \tan^{n-2} x \cdot \tan^2 x \, dx \]
We know that \( \tan^2 x = \sec^2 x - 1 \), so we can rewrite the integral:
\[ I_n = \int_0^{\frac{\pi}{4}} \tan^{n-2} x \cdot (\sec^2 x - 1) \, dx \]
This gives us:
\[ I_n = \int_0^{\frac{\pi}{4}} \tan^{n-2} x \cdot \sec^2 x \, dx - \int_0^{\frac{\pi}{4}} \tan^{n-2} x \, dx \]

3. **Evaluating the First Integral**:
The first integral can be evaluated using the substitution \( u = \tan x \), which gives \( du = \sec^2 x \, dx \). Thus, we have:
\[ \int_0^{\frac{\pi}{4}} \tan^{n-2} x \cdot \sec^2 x \, dx = \int_0^1 u^{n-2} \, du = \frac{1}{n-1} \]

4. **Combining the Results**:
Therefore, we have:
\[ I_n = \frac{1}{n-1} - I_{n-2} \]

5. **Finding Relationships**:
We can find \( I_{n+2} \) in terms of \( I_n \):
\[ I_{n+2} = \frac{1}{n+1} - I_n \]

6. **Establishing the Recurrence**:
From the above, we can establish a recurrence relation:
\[ I_n + I_{n+2} = \frac{1}{n+1} \]

7. **Finding the Values**:
Now, we can compute:
- \( I_2 + I_4 = \frac{1}{3} \)
- \( I_3 + I_5 = \frac{1}{4} \)
- \( I_4 + I_6 = \frac{1}{5} \)

8. **Finding the Required Values**:
We need to find:
\[ \frac{1}{I_2 + I_4}, \frac{1}{I_3 + I_5}, \frac{1}{I_4 + I_6}, \ldots \]
This gives us:
- \( \frac{1}{I_2 + I_4} = 3 \)
- \( \frac{1}{I_3 + I_5} = 4 \)
- \( \frac{1}{I_4 + I_6} = 5 \)

9. **Identifying the Sequence**:
The sequence \( 3, 4, 5, \ldots \) is an arithmetic progression (AP) with a common difference of 1.

### Final Answer:
The sequence \( \frac{1}{I_2 + I_4}, \frac{1}{I_3 + I_5}, \frac{1}{I_4 + I_6}, \ldots \) forms an arithmetic progression.