If `a_(0)=x,a_(n+1)=f(a_(n)), " where " n=0,1,2, …,` then answer the following questions.
If `f:R to R ` is given by `f(x)=3+4x and a_(n)=A+Bx,` then which of the following is not true?

Options

A. `A+B+1=2^(2n+1)`
B. `|A-B|=1`
C. `underset (h to oo)(lim)(A)/(B)= -1`
D. None of these

Explanation

To solve the problem step by step, we will analyze the given function and the recursive sequence defined by it.

### Step 1: Define the initial conditions and function
We are given:
- \( a_0 = x \)
- \( a_{n+1} = f(a_n) \)
- The function \( f(x) = 3 + 4x \)

### Step 2: Calculate the first few terms of the sequence
1. **Calculate \( a_1 \)**:
\[ a_1 = f(a_0) = f(x) = 3 + 4x \]

2. **Calculate \( a_2 \)**:
\[ a_2 = f(a_1) = f(3 + 4x) = 3 + 4(3 + 4x) = 3 + 12 + 16x = 15 + 16x \]

3. **Calculate \( a_3 \)**:
\[ a_3 = f(a_2) = f(15 + 16x) = 3 + 4(15 + 16x) = 3 + 60 + 64x = 63 + 64x \]

### Step 3: Identify a pattern
From the calculations, we can observe:
- \( a_1 = 3 + 4x \)
- \( a_2 = 15 + 16x \)
- \( a_3 = 63 + 64x \)

We can see that:
- The constant part seems to follow the pattern \( 4^n - 1 \).
- The coefficient of \( x \) is \( 4^n \).

### Step 4: Generalize the formula
We can generalize the terms as:
\[ a_n = (4^n - 1) + 4^n x \]
Thus, we can express:
- \( A = 4^n - 1 \)
- \( B = 4^n \)

### Step 5: Analyze the options
Now we need to check which of the following statements is not true based on our derived formula:
1. \( A + B = 4^n - 1 + 4^n = 2 \cdot 4^n - 1 \)
2. \( |A - B| = |(4^n - 1) - 4^n| = |-1| = 1 \)

### Step 6: Conclusion
After checking the derived values and the conditions, we find that the statement which is not true is the one that contradicts our findings.

### Final Answer
The option that is not true is the one that states \( |A - B| = -1 \), as we derived that \( |A - B| = 1 \).