What is the next number in this sequence: $\dfrac12, 2, 5, 11, 23, 47, 95, \ldots$?

$$\small{\text{ What is the next number in this sequence: $\dfrac12, 2, 5, 11, 23, 47, 95, \ldots$? }}$$

$$\small{\text{$ t_1 = \dfrac{1}{2} $}}\\\\ \small{\text{$ \begin{array}{rcl} t_n &=& 2^{n-1}t_1 +\sum \limits_{k=0}^{n-2}2^k \qquad | \qquad \sum \limits_{k=0}^{n-2}2^k = 2^{n-1}-1 \\\\ t_n &=& 2^{n-1}t_1 + 2^{n-1}-1 \\\\ t_n &=& 2^{n-1}(1+t_1)-1 \\\\ t_n &=& 2^{n-1}\left(1+\dfrac{1}{2}\right)-1 \\\\ t_n &=& 2^{n-1}\left(\dfrac{3}{2}\right)-1 \\\\ \mathbf{t_n} & \mathbf{=} & \mathbf{3\cdot 2^{n-2}-1} \\\\ t_8 &=& 3\cdot 2^6-1\\\\ t_8 &=& 3\cdot 64 - 1 \\\\ \mathbf{t_8} & \mathbf{=} & \mathbf{191} \end{array} $}}$$

1/2 , 2, 5, 11, 23, 47, 95

The next term is found by doubling the previous term and adding 1  =   2(95) + 1   = 191