The problem gives us a recursive formula for a sequence $U_n$: $U_1 = 7$ $U_n = 3U_{n-1} + 2n$ for $n \ge 2$. We are asked to find the values of $U_2$, $U_3$, and $U_4$.

1. Problem Description

The problem gives us a recursive formula for a sequence $U_n$:

$U_1 = 7$

$U_n = 3U_{n-1} + 2n$ for $n \ge 2$.

We are asked to find the values of $U_2$, $U_3$, and $U_4$.

2. Solution Steps

First, we find $U_2$. Using the formula with $n=2$, we have:

$U_2 = 3U_{2-1} + 2(2) = 3U_1 + 4$

Since $U_1 = 7$, we get:

$U_2 = 3(7) + 4 = 21 + 4 = 25$.

Next, we find $U_3$. Using the formula with $n=3$, we have:

$U_3 = 3U_{3-1} + 2(3) = 3U_2 + 6$

Since $U_2 = 25$, we get:

$U_3 = 3(25) + 6 = 75 + 6 = 81$.

Finally, we find $U_4$. Using the formula with $n=4$, we have:

$U_4 = 3U_{4-1} + 2(4) = 3U_3 + 8$

Since $U_3 = 81$, we get:

$U_4 = 3(81) + 8 = 243 + 8 = 251$.

3. Final Answer

$U_2 = 25$

$U_3 = 81$

$U_4 = 251$