To solve the problem, we need to analyze the sequence defined by \( u_n = \int_0^{\frac{\pi}{4}} \tan^n x \, dx \) and determine whether the sums \( u_2 + u_4, u_3 + u_5, u_4 + u_6, \ldots \) form an arithmetic progression (AP), geometric progression (GP), harmonic progression (HP), or none of these.
### Step 1: Finding \( u_n \)
We start with the expression for \( u_n \):
\[
u_n = \int_0^{\frac{\pi}{4}} \tan^n x \, dx
\]
Using integration by parts or a known result, we can derive a relationship between \( u_n \) and \( u_{n-2} \):
\[
u_n + u_{n-2} = \frac{1}{n-1} \quad \text{for } n \geq 2
\]
### Step 2: Finding specific sums
We will compute the sums \( u_2 + u_4, u_3 + u_5, u_4 + u_6 \) using the relationship we derived.
1. **For \( n = 4 \)**:
\[
u_4 + u_2 = \frac{1}{4 - 1} = \frac{1}{3}
\]
2. **For \( n = 5 \)**:
\[
u_5 + u_3 = \frac{1}{5 - 1} = \frac{1}{4}
\]
3. **For \( n = 6 \)**:
\[
u_6 + u_4 = \frac{1}{6 - 1} = \frac{1}{5}
\]
### Step 3: Analyzing the sums
Now we have the following sums:
- \( u_2 + u_4 = \frac{1}{3} \)
- \( u_3 + u_5 = \frac{1}{4} \)
- \( u_4 + u_6 = \frac{1}{5} \)
### Step 4: Checking for Progression
To check if these sums form an arithmetic progression (AP), we need to check if the difference between consecutive terms is constant:
- The difference between the first two sums:
\[
\frac{1}{4} - \frac{1}{3} = \frac{3 - 4}{12} = -\frac{1}{12}
\]
- The difference between the second and third sums:
\[
\frac{1}{5} - \frac{1}{4} = \frac{4 - 5}{20} = -\frac{1}{20}
\]
Since the differences are not equal, the sums do not form an AP.
### Step 5: Checking for Harmonic Progression (HP)
To check if the sums form a harmonic progression, we need to check if the reciprocals of these sums are in arithmetic progression:
- The reciprocals are:
\[
3, 4, 5
\]
These numbers are in AP since:
\[
4 - 3 = 1 \quad \text{and} \quad 5 - 4 = 1
\]
### Conclusion
Since the reciprocals of the sums \( u_2 + u_4, u_3 + u_5, u_4 + u_6 \) are in arithmetic progression, we conclude that the original sums are in harmonic progression (HP).
Thus, the final answer is that \( u_2 + u_4, u_3 + u_5, u_4 + u_6, \ldots \) are in HP.
