Let `a_(n)` denote the number of all n-digit numbers formed by the digits 0,1 or both such that no consecutive digits in them are 0. Let `b_(n)` be the number of such n-digit integers ending with digit 1 and let `c_(n)` be the number of such n-digit integers ending with digit 0. Which of the following is correct ?

Options

A. `a_(17)=a_(16)+a_(15)`
B. `c_(17) ne c_(16)+c_(15)`
C. `b_(17) ne b_(16)+c _(16)`
D. `a_(17)=c_(17)+b_(16)`

Explanation

To solve the problem, we need to establish relationships between the sequences \( a_n \), \( b_n \), and \( c_n \) based on the definitions provided.

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

### Step 2: Establish the relationship between \( a_n \), \( b_n \), and \( c_n \)
From the definitions, we can express \( a_n \) as:
\[ a_n = b_n + c_n \]
This is because every n-digit number either ends with 1 (counted by \( b_n \)) or ends with 0 (counted by \( c_n \)).

### Step 3: Find expressions for \( b_n \) and \( c_n \)
1. **For \( b_n \)**: An n-digit number ending with 1 can be formed by taking any valid (n-1)-digit number (which can end with either 0 or 1) and appending a 1. Thus:
\[ b_n = a_{n-1} \]

2. **For \( c_n \)**: An n-digit number ending with 0 must have its previous digit as 1 (to avoid consecutive 0s). Therefore, it can be formed by taking any valid (n-1)-digit number that ends with 1 and appending a 0. Thus:
\[ c_n = b_{n-1} \]

### Step 4: Substitute \( b_n \) into the equation for \( c_n \)
Using \( b_n = a_{n-1} \):
\[ c_n = b_{n-1} = a_{n-2} \]

### Step 5: Substitute \( b_n \) and \( c_n \) back into \( a_n \)
Now we can express \( a_n \) in terms of \( a_{n-1} \) and \( a_{n-2} \):
\[ a_n = b_n + c_n = a_{n-1} + a_{n-2} \]

### Step 6: Establish the recurrence relation
Thus, we have established the recurrence relation:
\[ a_n = a_{n-1} + a_{n-2} \]
This is similar to the Fibonacci sequence.

### Step 7: Initial conditions
To fully define the sequence, we need initial conditions:
- For \( n = 1 \): The valid 1-digit numbers are 0 and 1, so \( a_1 = 2 \).
- For \( n = 2 \): The valid 2-digit numbers are 01, 10, and 11 (not 00), so \( a_2 = 3 \).

### Conclusion
Now we can compute \( a_n \) for any \( n \) using the recurrence relation and initial conditions.