Let `a_n` denote the number of all n-digit positive integers formed by the digits 0, 1, or both such that no consecutive digits in them are 0. Let `b_n=`The number of such n-digit integers ending with digit 1 and `c_n=` Then number of such n-digit integers with digit 0.
The value of `b_6` is

Options

A. 7
B. 8
C. 9
D. 11

Answer

To solve the problem of finding \( b_6 \), we will first define our sequences and establish the necessary relationships.

### Step 1: Define the sequences
- Let \( a_n \) be the total number of n-digit integers formed by the digits 0 and 1, such that no two consecutive digits are 0.
- Let \( b_n \) be the number of n-digit integers that end with the digit 1.
- Let \( c_n \) be the number of n-digit integers that end with the digit 0.

### Step 2: Establish relationships
From the problem, we can derive the following relationships:
1. \( a_n = b_n + c_n \)
2. \( b_n = b_{n-1} + c_{n-1} \) (An n-digit number ending in 1 can be formed by adding 1 to either an (n-1)-digit number ending in 1 or 0)
3. \( c_n = b_{n-1} \) (An n-digit number ending in 0 can only be formed by adding 0 to an (n-1)-digit number ending in 1)

### Step 3: Calculate initial values
Now, we will calculate the initial values for \( b_n \) and \( c_n \):
- For \( n = 1 \):
- \( b_1 = 1 \) (the only 1-digit number is 1)
- \( c_1 = 0 \) (no 1-digit number ends with 0)

- For \( n = 2 \):
- Possible numbers: 10, 11
- \( b_2 = 2 \) (10, 11)
- \( c_2 = 1 \) (10)

- For \( n = 3 \):
- Possible numbers: 101, 111, 110
- \( b_3 = 3 \) (101, 111, 110)
- \( c_3 = 2 \) (110, 111)

Now we can summarize:
- \( b_1 = 1, c_1 = 0 \)
- \( b_2 = 2, c_2 = 1 \)
- \( b_3 = 3, c_3 = 2 \)

### Step 4: Calculate further values
Now we can calculate \( b_4, b_5, \) and \( b_6 \) using the established relationships:
- For \( n = 4 \):
- \( c_4 = b_3 = 3 \)
- \( b_4 = b_3 + c_3 = 3 + 2 = 5 \)

- For \( n = 5 \):
- \( c_5 = b_4 = 5 \)
- \( b_5 = b_4 + c_4 = 5 + 3 = 8 \)

- For \( n = 6 \):
- \( c_6 = b_5 = 8 \)
- \( b_6 = b_5 + c_5 = 8 + 5 = 13 \)

### Final Result
Thus, the value of \( b_6 \) is \( 13 \).