A sequence is defined by $$ U_{n+1}=3 U_{n}+1 $$ (i) Write down the first four terms of the sequence, given that $$ U_{1}=0 $$. (ii) Hence describe the behaviour of the sequence. (iii) Express $$ U_{3} $$ in terms of $$ U_{1} $$.

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To solve the problem, we will follow the steps outlined in the question.

### (i) Write down the first four terms of the sequence, given that $$ U_{1}=0 $$.

We start with the initial term $$ U_1 = 0 $$ and use the recurrence relation $$ U_{n+1} = 3U_n + 1 $$ to find the subsequent terms.

1. **Calculate $$ U_2 $$**:
   $$
   U_2 = 3U_1 + 1 = 3 \cdot 0 + 1 = 1
   $$

2. **Calculate $$ U_3 $$**:
   $$
   U_3 = 3U_2 + 1 = 3 \cdot 1 + 1 = 3 + 1 = 4
   $$

3. **Calculate $$ U_4 $$**:
   $$
   U_4 = 3U_3 + 1 = 3 \cdot 4 + 1 = 12 + 1 = 13
   $$

Thus, the first four terms of the sequence are:
$$
U_1 = 0, \quad U_2 = 1, \quad U_3 = 4, \quad U_4 = 13
$$

### (ii) Hence describe the behaviour of the sequence.

To analyze the behavior of the sequence, we can observe the terms we calculated:
- $$ U_1 = 0 $$
- $$ U_2 = 1 $$
- $$ U_3 = 4 $$
- $$ U_4 = 13 $$

We can see that the sequence is increasing. To understand the growth rate, we can look at the recurrence relation:
$$
U_{n+1} = 3U_n + 1
$$
This indicates that each term is approximately three times the previous term, plus one. As $$ n $$ increases, the "+1" becomes less significant compared to the multiplication by 3. Therefore, we can conclude that the sequence grows exponentially.

### (iii) Express $$ U_3 $$ in terms of $$ U_1 $$.

From our earlier calculations, we found:
$$
U_3 = 4
$$
Now, we can express $$ U_3 $$ in terms of $$ U_1 $$ using the recurrence relation. We know:
$$
U_2 = 3U_1 + 1
$$
Substituting $$ U_1 = 0 $$:
$$
U_2 = 3 \cdot 0 + 1 = 1
$$
Now, substituting $$ U_2 $$ into the equation for $$ U_3 $$:
$$
U_3 = 3U_2 + 1 = 3(3U_1 + 1) + 1 = 9U_1 + 3 + 1 = 9U_1 + 4
$$

Thus, we can express $$ U_3 $$ in terms of $$ U_1 $$ as:
$$
U_3 = 9U_1 + 4
$$

In summary:
- The first four terms are $$ 0, 1, 4, 13 $$.
- The sequence grows exponentially.
- $$ U_3 = 9U_1 + 4 $$.
The first four terms are 0, 1, 4, 13. The sequence grows exponentially. $$ U_3 = 9U_1 + 4 $$.

A sequence is defined by \( U_{n+1}=3 U_{n}+1 \) (i) Write down the first four terms of the sequence, given that \( U_{1}=0 \). (ii) Hence describe the behaviour of the sequence. (iii) Express \( U_{3} \) in terms of \( U_{1} \).

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