`(11)_(2)+(22)_(3)+(33)_(4)+(44)_(5)+(55)_(6)+(66)_(7)+(77)_(8)+(88)_(9)=(?)_(10)`

Options

A. 396
B. 276
C. 250
D. 342

Answer

To solve the problem, we need to convert each number from its respective base to base 10 and then sum them up. Let's break it down step by step.

### Step 1: Convert each number to base 10

1. **Convert \( (11)_2 \) to base 10**:
\[ (11)_2 = 1 \times 2^1 + 1 \times 2^0 = 2 + 1 = 3 \]

2. **Convert \( (22)_3 \) to base 10**:
\[ (22)_3 = 2 \times 3^1 + 2 \times 3^0 = 6 + 2 = 8 \]

3. **Convert \( (33)_4 \) to base 10**:
\[ (33)_4 = 3 \times 4^1 + 3 \times 4^0 = 12 + 3 = 15 \]

4. **Convert \( (44)_5 \) to base 10**:
\[ (44)_5 = 4 \times 5^1 + 4 \times 5^0 = 20 + 4 = 24 \]

5. **Convert \( (55)_6 \) to base 10**:
\[ (55)_6 = 5 \times 6^1 + 5 \times 6^0 = 30 + 5 = 35 \]

6. **Convert \( (66)_7 \) to base 10**:
\[ (66)_7 = 6 \times 7^1 + 6 \times 7^0 = 42 + 6 = 48 \]

7. **Convert \( (77)_8 \) to base 10**:
\[ (77)_8 = 7 \times 8^1 + 7 \times 8^0 = 56 + 7 = 63 \]

8. **Convert \( (88)_9 \) to base 10**:
\[ (88)_9 = 8 \times 9^1 + 8 \times 9^0 = 72 + 8 = 80 \]

### Step 2: Sum all the converted values

Now we will sum all the values we calculated:

\[ 3 + 8 + 15 + 24 + 35 + 48 + 63 + 80 \]

Calculating step by step:

- \( 3 + 8 = 11 \)
- \( 11 + 15 = 26 \)
- \( 26 + 24 = 50 \)
- \( 50 + 35 = 85 \)
- \( 85 + 48 = 133 \)
- \( 133 + 63 = 196 \)
- \( 196 + 80 = 276 \)

### Final Answer

Thus, the final answer is:
\[ (11)_{2} + (22)_{3} + (33)_{4} + (44)_{5} + (55)_{6} + (66)_{7} + (77)_{8} + (88)_{9} = (276)_{10} \]