Question:

Step 1: Identify the initial conditions for the sequence. The sequence is defined with the following initial information: For $a_0$, there are $2$ outcomes specified as $\{H, T\}$. For $a_2$, there are $3$ outcomes specified as $\{HT, TH, TT\}$. Step 2: Establish the recurrence relation for the sequence. For $n \geq 3$, the method for constructing sequences of length $n$ is given based on the first outcome: 1. If the first outcome is $H$, it must be followed by $T$. The remaining $n-2$ outcomes form a valid sequence, contributing $a_{n-2}$ possibilities. 2. If the first outcome is $T$, any valid sequence of length $n-1$ can follow it. This contributes $a_{n-1}$ possibilities. Combining these two cases, the recurrence relation for the sequence is established as: $$ a_n = a_{n-1} + a_{n-2} $$ Step 3: Calculate subsequent terms using the recurrence relation. To apply the recurrence $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$, we need two consecutive initial terms. From the descriptions in Step 1: The phrase "$a_0 = 2$ with outcomes in $\{H, T\}$" implies that for sequences of length 1, there are 2 possibilities (H or T), thus suggesting $a_1 = 2$. The phrase "$a_2 = 3$ with outcomes in $\{HT, TH, TT\}$" directly provides $a_2 = 3$. Using these effective initial conditions $a_1 = 2$ and $a_2 = 3$: We calculate $a_3$: $$ a_3 = a_2 + a_1 = 3 + 2 = 5 $$ Next, we calculate $a_4$: $$ a_4 = a_3 + a_2 = 5 + 3 = 8 $$ The final answer consists of the calculated terms: $a_3 = 5$ and $a_4 = 8$.