Question
The sequence \( u_{1}, u_{2}, u_{3}, \cdots \) is given by the recurrence relation \[ u_{n+1}=3 u_{n}+2, \quad u_{1}=2 \text {. } \] Find \( u_{2}, u_{3}, u_{4} \) and \( u_{5} \).
Ask by Crawford Barker. in Malawi
Feb 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( u_{2} = 8 \), \( u_{3} = 26 \), \( u_{4} = 80 \), \( u_{5} = 242 \).
Solution
We are given the recurrence relation:
uₙ₊₁ = 3uₙ + 2 with u₁ = 2.
Step 1: Find u₂.
u₂ = 3u₁ + 2 = 3(2) + 2 = 6 + 2 = 8
Step 2: Find u₃.
u₃ = 3u₂ + 2 = 3(8) + 2 = 24 + 2 = 26
Step 3: Find u₄.
u₄ = 3u₃ + 2 = 3(26) + 2 = 78 + 2 = 80
Step 4: Find u₅.
u₅ = 3u₄ + 2 = 3(80) + 2 = 240 + 2 = 242
Thus, the values are:
u₂ = 8,
u₃ = 26,
u₄ = 80,
u₅ = 242.
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Simplify this solution Beyond the Answer
To find \( u_2, u_3, u_4, \) and \( u_5 \), we will use the recurrence relation \( u_{n+1} = 3u_n + 2 \) starting from the initial condition \( u_1 = 2 \). 1. **Calculate \( u_2 \)**: \[ u_2 = 3u_1 + 2 = 3 \times 2 + 2 = 6 + 2 = 8 \] 2. **Calculate \( u_3 \)**: \[ u_3 = 3u_2 + 2 = 3 \times 8 + 2 = 24 + 2 = 26 \] 3. **Calculate \( u_4 \)**: \[ u_4 = 3u_3 + 2 = 3 \times 26 + 2 = 78 + 2 = 80 \] 4. **Calculate \( u_5 \)**: \[ u_5 = 3u_4 + 2 = 3 \times 80 + 2 = 240 + 2 = 242 \] Thus, the values are: \[ u_2 = 8, \quad u_3 = 26, \quad u_4 = 80, \quad u_5 = 242. \]
