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.
`f(9235)+f(9450)` equals:

Options

A. 5
B. 3
C. 2
D. none of these

Answer

To solve the problem, we need to evaluate the function \( f(a_1, a_2, a_3, \ldots, a_n) \) for the numbers 9235 and 9450, and then sum the results. The function is defined as:

\[ f(a_1, a_2, a_3, \ldots, a_n) = a_1 \cdot 2^{n-1} + a_2 \cdot 2^{n-2} + a_3 \cdot 2^{n-3} + \ldots + a_n \cdot 2^{0} \]

where \( n \) is the number of digits in the number.

### Step 1: Calculate \( f(9235) \)

1. Identify the digits:
- \( a_1 = 9 \)
- \( a_2 = 2 \)
- \( a_3 = 3 \)
- \( a_4 = 5 \)
- \( n = 4 \) (since there are 4 digits)

2. Apply the function:
\[ f(9235) = 9 \cdot 2^{4-1} + 2 \cdot 2^{4-2} + 3 \cdot 2^{4-3} + 5 \cdot 2^{4-4} \]
\[ = 9 \cdot 2^3 + 2 \cdot 2^2 + 3 \cdot 2^1 + 5 \cdot 2^0 \]
\[ = 9 \cdot 8 + 2 \cdot 4 + 3 \cdot 2 + 5 \cdot 1 \]
\[ = 72 + 8 + 6 + 5 = 91 \]

### Step 2: Reduce \( 91 \) to a single digit

1. Calculate \( f(91) \):
- \( a_1 = 9 \)
- \( a_2 = 1 \)
- \( n = 2 \)

2. Apply the function:
\[ f(91) = 9 \cdot 2^{2-1} + 1 \cdot 2^{2-2} \]
\[ = 9 \cdot 2^1 + 1 \cdot 2^0 \]
\[ = 9 \cdot 2 + 1 \cdot 1 \]
\[ = 18 + 1 = 19 \]

3. Calculate \( f(19) \):
- \( a_1 = 1 \)
- \( a_2 = 9 \)
- \( n = 2 \)

4. Apply the function:
\[ f(19) = 1 \cdot 2^{2-1} + 9 \cdot 2^{2-2} \]
\[ = 1 \cdot 2 + 9 \cdot 1 \]
\[ = 2 + 9 = 11 \]

5. Calculate \( f(11) \):
- \( a_1 = 1 \)
- \( a_2 = 1 \)
- \( n = 2 \)

6. Apply the function:
\[ f(11) = 1 \cdot 2^{2-1} + 1 \cdot 2^{2-2} \]
\[ = 1 \cdot 2 + 1 \cdot 1 \]
\[ = 2 + 1 = 3 \]

So, \( f(9235) = 3 \).

### Step 3: Calculate \( f(9450) \)

1. Identify the digits:
- \( a_1 = 9 \)
- \( a_2 = 4 \)
- \( a_3 = 5 \)
- \( a_4 = 0 \)
- \( n = 4 \)

2. Apply the function:
\[ f(9450) = 9 \cdot 2^{4-1} + 4 \cdot 2^{4-2} + 5 \cdot 2^{4-3} + 0 \cdot 2^{4-4} \]
\[ = 9 \cdot 8 + 4 \cdot 4 + 5 \cdot 2 + 0 \cdot 1 \]
\[ = 72 + 16 + 10 + 0 = 98 \]

### Step 4: Reduce \( 98 \) to a single digit

1. Calculate \( f(98) \):
- \( a_1 = 9 \)
- \( a_2 = 8 \)
- \( n = 2 \)

2. Apply the function:
\[ f(98) = 9 \cdot 2^{2-1} + 8 \cdot 2^{2-2} \]
\[ = 9 \cdot 2 + 8 \cdot 1 \]
\[ = 18 + 8 = 26 \]

3. Calculate \( f(26) \):
- \( a_1 = 2 \)
- \( a_2 = 6 \)
- \( n = 2 \)

4. Apply the function:
\[ f(26) = 2 \cdot 2^{2-1} + 6 \cdot 2^{2-2} \]
\[ = 2 \cdot 2 + 6 \cdot 1 \]
\[ = 4 + 6 = 10 \]

5. Calculate \( f(10) \):
- \( a_1 = 1 \)
- \( a_2 = 0 \)
- \( n = 2 \)

6. Apply the function:
\[ f(10) = 1 \cdot 2^{2-1} + 0 \cdot 2^{2-2} \]
\[ = 1 \cdot 2 + 0 \cdot 1 \]
\[ = 2 + 0 = 2 \]

So, \( f(9450) = 2 \).

### Step 5: Final Calculation

Now we sum the results:
\[ f(9235) + f(9450) = 3 + 2 = 5 \]

### Final Answer:
\[ \boxed{5} \]