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 \( U_{1} \).

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. - \( U_3 = 9U_1 + 4 \).