The problem asks us to find the next three terms of the recursive sequence defined by $a_n = 2a_{n-1} - 1$ for $n \ge 2$, with the initial term $a_1 = 3$.

1. Problem Description

The problem asks us to find the next three terms of the recursive sequence defined by $a_n = 2a_{n-1} - 1$ for $n \ge 2$, with the initial term $a_1 = 3$.

2. Solution Steps

We are given the recursive formula $a_n = 2a_{n-1} - 1$ and $a_1 = 3$. We need to find $a_2, a_3,$ and $a_4$.

For $n=2$:

$a_2 = 2a_{2-1} - 1 = 2a_1 - 1 = 2(3) - 1 = 6 - 1 = 5$.

For $n=3$:

$a_3 = 2a_{3-1} - 1 = 2a_2 - 1 = 2(5) - 1 = 10 - 1 = 9$.

For $n=4$:

$a_4 = 2a_{4-1} - 1 = 2a_3 - 1 = 2(9) - 1 = 18 - 1 = 17$.

3. Final Answer

The next three terms are 5, 9, and 17.