Solve the following questions on the basis of following functions
`P_(n+1) = P_(n)-P_(n-1)`, P is the term of the sequence and `P_(0) = 0, P_(1) =1`
`Q_(n+1) = Q_(n) +Q_(n-1)`, Q is the term of the sequence and `Q_(0) = 0, Q_(1) = 1`
Which of the following can be a positive value of `Q_n+P_(n+1)`?

Options

A. 55
B. 234
C. 146
D. none of these

Answer

To solve the problem, we need to analyze the sequences defined by the functions given and then find the possible positive values of \( Q_n + P_{n+1} \).

### Step 1: Define the sequences \( P_n \) and \( Q_n \)

1. **For the sequence \( P_n \)**:
- The recurrence relation is given by:
\[ P_{n+1} = P_n - P_{n-1} \]
- The initial conditions are:
\[ P_0 = 0, \quad P_1 = 1 \]

2. **For the sequence \( Q_n \)**:
- The recurrence relation is given by:
\[ Q_{n+1} = Q_n + Q_{n-1} \]
- The initial conditions are:
\[ Q_0 = 0, \quad Q_1 = 1 \]

### Step 2: Calculate the terms of the sequences

1. **Calculating \( P_n \)**:
- \( P_2 = P_1 - P_0 = 1 - 0 = 1 \)
- \( P_3 = P_2 - P_1 = 1 - 1 = 0 \)
- \( P_4 = P_3 - P_2 = 0 - 1 = -1 \)
- \( P_5 = P_4 - P_3 = -1 - 0 = -1 \)
- \( P_6 = P_5 - P_4 = -1 - (-1) = 0 \)
- Continuing this process, we find:
\[ P_0 = 0, \quad P_1 = 1, \quad P_2 = 1, \quad P_3 = 0, \quad P_4 = -1, \quad P_5 = -1, \quad P_6 = 0, \quad P_7 = 1, \quad P_8 = 1, \quad P_9 = 0, \quad P_{10} = -1, \quad P_{11} = -1, \ldots \]

2. **Calculating \( Q_n \)**:
- \( Q_2 = Q_1 + Q_0 = 1 + 0 = 1 \)
- \( Q_3 = Q_2 + Q_1 = 1 + 1 = 2 \)
- \( Q_4 = Q_3 + Q_2 = 2 + 1 = 3 \)
- \( Q_5 = Q_4 + Q_3 = 3 + 2 = 5 \)
- \( Q_6 = Q_5 + Q_4 = 5 + 3 = 8 \)
- Continuing this process, we find:
\[ Q_0 = 0, \quad Q_1 = 1, \quad Q_2 = 1, \quad Q_3 = 2, \quad Q_4 = 3, \quad Q_5 = 5, \quad Q_6 = 8, \quad Q_7 = 13, \quad Q_8 = 21, \quad Q_9 = 34, \quad Q_{10} = 55, \quad Q_{11} = 89, \ldots \]

### Step 3: Calculate \( Q_n + P_{n+1} \)

Now, we will calculate \( Q_n + P_{n+1} \) for various values of \( n \):

- For \( n = 0 \):
\[ Q_0 + P_1 = 0 + 1 = 1 \]
- For \( n = 1 \):
\[ Q_1 + P_2 = 1 + 1 = 2 \]
- For \( n = 2 \):
\[ Q_2 + P_3 = 1 + 0 = 1 \]
- For \( n = 3 \):
\[ Q_3 + P_4 = 2 - 1 = 1 \]
- For \( n = 4 \):
\[ Q_4 + P_5 = 3 - 1 = 2 \]
- For \( n = 5 \):
\[ Q_5 + P_6 = 5 + 0 = 5 \]
- For \( n = 6 \):
\[ Q_6 + P_7 = 8 + 1 = 9 \]
- For \( n = 7 \):
\[ Q_7 + P_8 = 13 + 1 = 14 \]
- For \( n = 8 \):
\[ Q_8 + P_9 = 21 + 0 = 21 \]
- For \( n = 9 \):
\[ Q_9 + P_{10} = 34 - 1 = 33 \]
- For \( n = 10 \):
\[ Q_{10} + P_{11} = 55 - 1 = 54 \]
- For \( n = 11 \):
\[ Q_{11} + P_{12} = 89 + 0 = 89 \]

### Step 4: Identify possible positive values

From the calculations, we see that the possible positive values of \( Q_n + P_{n+1} \) include:
- 1, 2, 5, 9, 14, 21, 33, 54, 89, etc.

### Conclusion

Now we can check which of the provided options (55, 34, 146, etc.) can be a positive value of \( Q_n + P_{n+1} \).

- **None of the options match the calculated values.**

Thus, the answer is **none of these**.