A binary sequence is a sequence in which each term is equal to $0$ or $1$. A binary sequence is called $\text{friendly}$ if each term is adjacent to at least on term that is equal to $1$. For example , the sequence $0,1,1,0,0,1,1,1$ is $\text{friendly}$. Let $F_{n}$ denote the number of $\text{friendly}$ binary sequences with $n$ terms. Find the smallest positive integer $n\ge 2$ such that $F_{n}>100$
1. **Define the problem and notation:**
   - A binary sequence is a sequence where each term is either \(0\) or \(1\).
   - A binary sequence is called \(\text{friendly}\) if each term is adjacent to at least one term that is equal to \(1\).
   - Let \(F_n\) denote the number of \(\text{friendly}\) binary sequences with \(n\) terms.
   - We need to find the smallest positive integer \(n \geq 2\) such that \(F_n > 100\).

2. **Define subproblems:**
   - Let \(00_n\) denote the number of \(\text{friendly}\) sequences of length \(n\) starting with \(0\) and ending with \(0\).
   - Let \(01_n\) denote the number of \(\text{friendly}\) sequences of length \(n\) starting with \(0\) and ending with \(1\).
   - Let \(10_n\) denote the number of \(\text{friendly}\) sequences of length \(n\) starting with \(1\) and ending with \(0\).
   - Let \(11_n\) denote the number of \(\text{friendly}\) sequences of length \(n\) starting with \(1\) and ending with \(1\).

3. **Establish recursive formulas:**
   - **Formula 1:** \(00_n = 11_{n-2}\)
     - Proof: A sequence of type \(00\) must be of the form \(011...110\). Removing the first and last \(0\) gives a sequence of type \(11\) of length \(n-2\).
   - **Formula 2:** \(10_n = 11_{n-1}\)
     - Proof: A sequence of type \(10\) must be of the form \(11...110\). Removing the last \(0\) gives a sequence of type \(11\) of length \(n-1\).
   - **Formula 3:** \(01_n = 11_{n-1}\)
     - Proof: Similar to the argument for \(10_n\).
   - **Formula 4:** \(11_n = 11_{n-1} + 11_{n-3} + 11_{n-4}\)
     - Proof: Consider the sequence of type \(11\):
       - If the third member is \(1\), the sequence is of the form \(111...11\). Removing the first \(1\) gives a sequence of type \(11\) of length \(n-1\).
       - If the third member is \(0\) and the fourth member is \(1\), the sequence is of the form \(11011...11\). Removing the first three elements gives a sequence of type \(11\) of length \(n-3\).
       - If the third member is \(0\) and the fourth member is \(0\), the sequence is of the form \(110011...11\). Removing the first four elements gives a sequence of type \(11\) of length \(n-4\).

4. **Calculate initial values:**
   - For \(n = 1\): \(00_1 = 0\), \(01_1 = 0\), \(10_1 = 0\), \(11_1 = 0\), \(F_1 = 0\)
   - For \(n = 2\): \(00_2 = 0\), \(01_2 = 0\), \(10_2 = 0\), \(11_2 = 1\), \(F_2 = 1\)

5. **Use the recursive formulas to compute \(F_n\) for larger \(n\):**
   - \(F_n = 00_n + 01_n + 10_n + 11_n\)

6. **Construct the table:**
   \[
   \begin{array}{|c|c|c|c|c|c|}
   \hline
   n & 00_n & 01_n & 10_n & 11_n & F_n \\
   \hline
   1 & 0 & 0 & 0 & 0 & 0 \\
   2 & 0 & 0 & 0 & 1 & 1 \\
   3 & 0 & 1 & 1 & 1 & 3 \\
   4 & 1 & 1 & 1 & 1 & 4 \\
   5 & 1 & 1 & 1 & 2 & 5 \\
   6 & 1 & 2 & 2 & 4 & 9 \\
   7 & 2 & 4 & 4 & 6 & 16 \\
   8 & 4 & 6 & 6 & 9 & 25 \\
   9 & 6 & 9 & 9 & 15 & 39 \\
   10 & 9 & 15 & 15 & 25 & 64 \\
   11 & 15 & 25 & 25 & 40 & 105 \\
   \hline
   \end{array}
   \]

7. **Conclusion:**
   - The smallest \(n\) for which \(F_n > 100\) is \(n = 11\).

The final answer is \( \boxed{ n = 11 } \)