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The problem states that we are dealing with an arithmetic sequence, and the following information is given:

1. \( U_3 + U_9 + U_{11} = 75 \),
2. The middle term of the sequence is 68,
3. The total number of terms is 43,
4. We are asked to find \( U_{43} \).

Let's break it down step by step.

### General Formula for Arithmetic Sequence:
For an arithmetic sequence, the general term \( U_n \) is given by:
\[
U_n = a + (n - 1) d
\]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 1: Middle Term Information
The middle term is given as 68. Since the total number of terms is 43, the middle term is:
\[
U_{\text{mid}} = U_{22}
\]
So, we know:
\[
U_{22} = 68
\]
Using the formula for \( U_{22} \):
\[
a + 21d = 68 \quad \text{(Equation 1)}
\]

### Step 2: Sum of \( U_3 + U_9 + U_{11} \)
Using the formula for the terms \( U_3 \), \( U_9 \), and \( U_{11} \):
\[
U_3 = a + 2d, \quad U_9 = a + 8d, \quad U_{11} = a + 10d
\]
We are told:
\[
U_3 + U_9 + U_{11} = 75
\]
Substitute the expressions for \( U_3 \), \( U_9 \), and \( U_{11} \):
\[
(a + 2d) + (a + 8d) + (a + 10d) = 75
\]
Simplifying:
\[
3a + 20d = 75 \quad \text{(Equation 2)}
\]

### Step 3: Solving the System of Equations
We now have two equations:
1. \( a + 21d = 68 \),
2. \( 3a + 20d = 75 \).

Let's solve these equations simultaneously to find \( a \) and \( d \).

I'll compute this next.The solution to the system of equations is:
- The first term \( a = 5 \),
- The common difference \( d = 3 \).

### Step 4: Finding \( U_{43} \)
Now, to find \( U_{43} \), we use the formula for the \( n \)-th term:
\[
U_{43} = a + (43 - 1) d = 5 + 42 \times 3 = 5 + 126 = 131
\]

Thus, \( U_{43} = 131 \).

Would you like more details on any part of the solution? Here are 5 related questions:

1. How would you find the first term if different conditions were given?
2. How does the middle term in an arithmetic sequence relate to the sum?
3. Can we determine the sum of all 43 terms using this information?
4. How would the sequence change if the number of terms increased?
5. Can we derive the formula for the sum of an arithmetic series?

**Tip:** When solving systems of equations in sequences, identifying known terms such as the middle or sum helps quickly find crucial sequence parameters.

Diketahui suatu barisan aritmatika dengan U3 + U9 + U11 = 75. Suku tengah barisan tersebut adalah 68 dan banyak sukunya 43. Tentukan U43.

Pelajari cara menghitung suku ke-43 (U43) dalam barisan aritmatika dengan suku tengah 68 dan jumlah sukunya 43. Gunakan formula umum barisan aritmatika dan sistem persamaan linier untuk menyelesaikan soal ini. Cocok untuk siswa SMA kelas 10-12.

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