Find the missing values A and B to balance the sums on each side of the triangle.

Let's solve each problem step by step.

---

Q1. Find out A and B so as to make the sum equal on each side of this triangle.

The triangle is structured as follows:
```
3
/ \
A B
/ \
2-------6-------1
```

We need to find values for \( A \) and \( B \) such that the sum of the numbers along each side of the triangle is equal.

#### Step 1: Identify the sides of the triangle
The three sides are:
1. \( 3 + A + 2 \)
2. \( 3 + B + 1 \)
3. \( 2 + 6 + 1 \)

#### Step 2: Calculate the sum of the third side
The third side is \( 2 + 6 + 1 = 9 \).

#### Step 3: Set up equations for the other two sides
For the first side:
\[ 3 + A + 2 = 9 \]
\[ 5 + A = 9 \]
\[ A = 4 \]

For the second side:
\[ 3 + B + 1 = 9 \]
\[ 4 + B = 9 \]
\[ B = 5 \]

#### Step 4: Verify the solution
- First side: \( 3 + 4 + 2 = 9 \)
- Second side: \( 3 + 5 + 1 = 9 \)
- Third side: \( 2 + 6 + 1 = 9 \)

All sides sum to 9, so the solution is correct.

Answer:
\[
\boxed{A = 4, B = 5}
\]

---

Q2. Study the pattern and fill in the missing number.

The sequence is:
\[ 4P, 3Q, 2R, ? \]

#### Step 1: Analyze the pattern
- The first part of each term is a decreasing sequence of numbers: \( 4, 3, 2, ? \).
- The second part of each term is a sequence of letters: \( P, Q, R, ? \).

#### Step 2: Determine the next number
- The numbers decrease by 1: \( 4, 3, 2, 1 \).
- The letters follow the alphabetical order: \( P, Q, R, S \).

#### Step 3: Combine the results
The next term is \( 1S \).

Answer:
\[
\boxed{1S}
\]

---

Q3. Fill in the missing number.

The sequence is:
\[ 3, 6, 12, ? \]

#### Step 1: Analyze the pattern
- Each term is multiplied by 2 to get the next term:
\[ 3 \times 2 = 6 \]
\[ 6 \times 2 = 12 \]

#### Step 2: Determine the next number
- The next term is:
\[ 12 \times 2 = 24 \]

Answer:
\[
\boxed{24}
\]

---

Q4. Fill in the missing number.

The sequence is:
\[ 80, 40, 20, ? \]

#### Step 1: Analyze the pattern
- Each term is divided by 2 to get the next term:
\[ 80 \div 2 = 40 \]
\[ 40 \div 2 = 20 \]

#### Step 2: Determine the next number
- The next term is:
\[ 20 \div 2 = 10 \]

Answer:
\[
\boxed{10}
\]

---

Q5. Fill in the missing number.

The sequence is:
\[ 6A7, 7B7, 8C7, ? \]

#### Step 1: Analyze the pattern
- The first digit increases by 1: \( 6, 7, 8, ? \).
- The middle letter follows the alphabetical order: \( A, B, C, ? \).
- The last digit remains constant: \( 7 \).

#### Step 2: Determine the next number
- The next digit is \( 9 \).
- The next letter is \( D \).
- The last digit remains \( 7 \).

#### Step 3: Combine the results
The next term is \( 9D7 \).

Answer:
\[
\boxed{9D7}
\]

---

Q6. Fill in the missing number.

The structure is:
```
30
/ \
20 ?
/ /
5 10 15
```

#### Step 1: Analyze the pattern
- The top number (30) is the sum of the two numbers below it.
- The left branch: \( 20 = 5 + 10 \).
- The right branch: Let the missing number be \( x \). Then:
\[ 30 = 20 + x \]

#### Step 2: Solve for \( x \)
\[ x = 30 - 20 = 10 \]

Answer:
\[
\boxed{10}
\]

---

Q7. Fill in the missing number.

The table is:
```
3 4 5 6
---------
9 15 25 ?
```

#### Step 1: Analyze the pattern
- Each number in the bottom row appears to be related to the corresponding number in the top row.
- For the first column:
\[ 3 \times 3 = 9 \]
- For the second column:
\[ 4 \times 3.75 = 15 \]
- For the third column:
\[ 5 \times 5 = 25 \]

#### Step 2: Identify the relationship
The pattern seems to be:
\[ \text{Bottom number} = \text{Top number} \times (\text{Top number} - 1) \]

#### Step 3: Apply the pattern to the fourth column
- For the fourth column:
\[ 6 \times (6 - 1) = 6 \times 5 = 30 \]

Answer:
\[
\boxed{30}
\]

---

Final Answers:

1. \(\boxed{A = 4, B = 5}\)
2. \(\boxed{1S}\)
3. \(\boxed{24}\)
4. \(\boxed{10}\)
5. \(\boxed{9D7}\)
6. \(\boxed{10}\)
7. \(\boxed{30}\)