A function is defined as follows :
`f(a_(1), a_(2), a_(3)...a_(n))=a_(1)2^(n-1)+a_(2)2^(n-2)+a_(3)2^(n-3)+...a_(n)2^(0)`
The above function is repreated until the value of function reduces to a single digit number.
What is the value of `f[f(888222)+f(113113)]:`

Options

A. 6
B. 7
C. 8
D. 9

Answer

To solve the problem, we will follow the steps outlined in the video transcript to evaluate the function \( f \) and reduce the results to a single digit.

### Step 1: Calculate \( f(888222) \)

1. **Identify \( n \)**: The number \( 888222 \) has 6 digits, so \( n = 6 \).
2. **Apply the function**:
\[ f(888222) = 8 \cdot 2^{5} + 8 \cdot 2^{4} + 8 \cdot 2^{3} + 2 \cdot 2^{2} + 2 \cdot 2^{1} + 2 \cdot 2^{0} \]
3. **Calculate each term**:
- \( 8 \cdot 2^{5} = 8 \cdot 32 = 256 \)
- \( 8 \cdot 2^{4} = 8 \cdot 16 = 128 \)
- \( 8 \cdot 2^{3} = 8 \cdot 8 = 64 \)
- \( 2 \cdot 2^{2} = 2 \cdot 4 = 8 \)
- \( 2 \cdot 2^{1} = 2 \cdot 2 = 4 \)
- \( 2 \cdot 2^{0} = 2 \cdot 1 = 2 \)
4. **Sum the results**:
\[ 256 + 128 + 64 + 8 + 4 + 2 = 462 \]

### Step 2: Reduce \( 462 \) to a single digit

1. **Identify \( n \)**: The number \( 462 \) has 3 digits, so \( n = 3 \).
2. **Apply the function**:
\[ f(462) = 4 \cdot 2^{2} + 6 \cdot 2^{1} + 2 \cdot 2^{0} \]
3. **Calculate each term**:
- \( 4 \cdot 2^{2} = 4 \cdot 4 = 16 \)
- \( 6 \cdot 2^{1} = 6 \cdot 2 = 12 \)
- \( 2 \cdot 2^{0} = 2 \cdot 1 = 2 \)
4. **Sum the results**:
\[ 16 + 12 + 2 = 30 \]

### Step 3: Reduce \( 30 \) to a single digit

1. **Identify \( n \)**: The number \( 30 \) has 2 digits, so \( n = 2 \).
2. **Apply the function**:
\[ f(30) = 3 \cdot 2^{1} + 0 \cdot 2^{0} \]
3. **Calculate each term**:
- \( 3 \cdot 2^{1} = 3 \cdot 2 = 6 \)
- \( 0 \cdot 2^{0} = 0 \)
4. **Sum the results**:
\[ 6 + 0 = 6 \]

### Step 4: Calculate \( f(113113) \)

1. **Identify \( n \)**: The number \( 113113 \) has 6 digits, so \( n = 6 \).
2. **Apply the function**:
\[ f(113113) = 1 \cdot 2^{5} + 1 \cdot 2^{4} + 3 \cdot 2^{3} + 1 \cdot 2^{2} + 1 \cdot 2^{1} + 3 \cdot 2^{0} \]
3. **Calculate each term**:
- \( 1 \cdot 2^{5} = 32 \)
- \( 1 \cdot 2^{4} = 16 \)
- \( 3 \cdot 2^{3} = 24 \)
- \( 1 \cdot 2^{2} = 4 \)
- \( 1 \cdot 2^{1} = 2 \)
- \( 3 \cdot 2^{0} = 3 \)
4. **Sum the results**:
\[ 32 + 16 + 24 + 4 + 2 + 3 = 81 \]

### Step 5: Reduce \( 81 \) to a single digit

1. **Identify \( n \)**: The number \( 81 \) has 2 digits, so \( n = 2 \).
2. **Apply the function**:
\[ f(81) = 8 \cdot 2^{1} + 1 \cdot 2^{0} \]
3. **Calculate each term**:
- \( 8 \cdot 2^{1} = 16 \)
- \( 1 \cdot 2^{0} = 1 \)
4. **Sum the results**:
\[ 16 + 1 = 17 \]

### Step 6: Reduce \( 17 \) to a single digit

1. **Identify \( n \)**: The number \( 17 \) has 2 digits, so \( n = 2 \).
2. **Apply the function**:
\[ f(17) = 1 \cdot 2^{1} + 7 \cdot 2^{0} \]
3. **Calculate each term**:
- \( 1 \cdot 2^{1} = 2 \)
- \( 7 \cdot 2^{0} = 7 \)
4. **Sum the results**:
\[ 2 + 7 = 9 \]

### Step 7: Final Calculation

Now we have:
- \( f(888222) = 6 \)
- \( f(113113) = 9 \)

Finally, we need to calculate:
\[ f[f(888222) + f(113113)] = f[6 + 9] = f[15] \]

### Step 8: Calculate \( f(15) \)

1. **Identify \( n \)**: The number \( 15 \) has 2 digits, so \( n = 2 \).
2. **Apply the function**:
\[ f(15) = 1 \cdot 2^{1} + 5 \cdot 2^{0} \]
3. **Calculate each term**:
- \( 1 \cdot 2^{1} = 2 \)
- \( 5 \cdot 2^{0} = 5 \)
4. **Sum the results**:
\[ 2 + 5 = 7 \]

### Final Answer
The value of \( f[f(888222) + f(113113)] \) is \( \boxed{7} \).