A fair coin is tossed `n` times. Let `a_(n)` denotes the number of cases in which no two heads occur consecutively. Then which of the following is not true ?

Options

A. `a_(1)=2`
B. `a_(2)=3`
C. `a_(5)=13`
D. `a_(8)=55`

Explanation

To solve the problem, we need to determine the number of ways to toss a fair coin `n` times such that no two heads occur consecutively. We denote this number as `a(n)`.

### Step-by-Step Solution:

1. **Understanding the Base Cases:**
- When the coin is tossed once (`n = 1`):
- Possible outcomes: H (Head) or T (Tail).
- Thus, `a(1) = 2`.

2. **Calculating for `n = 2`:**
- When the coin is tossed twice (`n = 2`):
- Possible outcomes: HH, HT, TH, TT.
- Valid outcomes (no two heads consecutively): HT, TH, TT.
- Thus, `a(2) = 3`.

3. **Finding a Recursive Formula:**
- For `n >= 3`, we can derive a recursive relation:
- If the first toss is H, the second must be T (to avoid consecutive heads), leaving us with `n-2` tosses. This gives us `a(n-2)` ways.
- If the first toss is T, the next toss can be either H or T, leaving us with `n-1` tosses. This gives us `a(n-1)` ways.
- Therefore, the recursive relation is:
\[ a(n) = a(n-1) + a(n-2) \]

4. **Calculating Further Values:**
- Using the recursive formula:
- For `n = 3`:
\[ a(3) = a(2) + a(1) = 3 + 2 = 5 \]
- For `n = 4`:
\[ a(4) = a(3) + a(2) = 5 + 3 = 8 \]
- For `n = 5`:
\[ a(5) = a(4) + a(3) = 8 + 5 = 13 \]
- For `n = 6`:
\[ a(6) = a(5) + a(4) = 13 + 8 = 21 \]
- For `n = 7`:
\[ a(7) = a(6) + a(5) = 21 + 13 = 34 \]
- For `n = 8`:
\[ a(8) = a(7) + a(6) = 34 + 21 = 55 \]

5. **Conclusion:**
- We have computed the values:
- \( a(1) = 2 \)
- \( a(2) = 3 \)
- \( a(3) = 5 \)
- \( a(4) = 8 \)
- \( a(5) = 13 \)
- \( a(6) = 21 \)
- \( a(7) = 34 \)
- \( a(8) = 55 \)

6. **Identifying the Incorrect Statement:**
- The question asks which of the following statements is not true. Based on our calculations:
- \( a(1) = 2 \) (True)
- \( a(2) = 3 \) (True)
- \( a(5) = 13 \) (True)
- \( a(8) = 55 \) (True)
- Therefore, all options provided in the question are correct.