Let f(n) denotes the number of different ways, the positive integer n ca be expressed as the sum of the 1's and 2's. for example, f(4)=5.
i.e., `4=1+1+1+1`
`=1+1+2=1+2+1=2+1+1=2+2`
Q. The number of solutions of the equation `f(n)=n`, where `n in N` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to find the number of solutions to the equation \( f(n) = n \), where \( f(n) \) is defined as the number of different ways to express the positive integer \( n \) as a sum of 1's and 2's.

### Step-by-Step Solution:

1. **Understanding \( f(n) \)**:
- The function \( f(n) \) counts the number of ways to express \( n \) using 1's and 2's.
- For example:
- \( f(1) = 1 \) (only \( 1 \))
- \( f(2) = 2 \) (either \( 1 + 1 \) or \( 2 \))
- \( f(3) = 3 \) (either \( 1 + 1 + 1 \), \( 1 + 2 \), or \( 2 + 1 \))
- \( f(4) = 5 \) (as given in the problem)

2. **Calculating \( f(n) \)** for \( n = 1, 2, 3, 4, 5 \):
- \( f(1) = 1 \)
- \( f(2) = 2 \)
- \( f(3) = 3 \)
- \( f(4) = 5 \)
- To find \( f(5) \):
- The combinations are:
- \( 1 + 1 + 1 + 1 + 1 \)
- \( 1 + 1 + 1 + 2 \) (and its permutations)
- \( 1 + 2 + 2 \) (and its permutations)
- Total combinations for \( f(5) \) = 8.

3. **Values of \( f(n) \)**:
- We summarize the values:
- \( f(1) = 1 \)
- \( f(2) = 2 \)
- \( f(3) = 3 \)
- \( f(4) = 5 \)
- \( f(5) = 8 \)

4. **Finding Solutions to \( f(n) = n \)**:
- We check each value of \( n \):
- For \( n = 1 \): \( f(1) = 1 \) (solution)
- For \( n = 2 \): \( f(2) = 2 \) (solution)
- For \( n = 3 \): \( f(3) = 3 \) (solution)
- For \( n = 4 \): \( f(4) = 5 \) (not a solution)
- For \( n = 5 \): \( f(5) = 8 \) (not a solution)

5. **Conclusion**:
- The solutions to the equation \( f(n) = n \) are \( n = 1, 2, 3 \).
- Therefore, the total number of solutions is **3**.

### Final Answer:
The number of solutions of the equation \( f(n) = n \) is **3**.

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