$T(n) = 2T(n-1) -1$
$\qquad =2(2T(n-2)-1) -1$
$\qquad =2^2T(n-2) -2 -1$
$\qquad =2^2(2T(n-3)-1) -2 -1$
$\qquad =2^3T(n-3) - 2^2 -2 -1$
$\qquad \dots $
$\qquad =2^{n-1}T(n-(n-1)) - \left(2^{n-2} +2^{n-3}+\dots+2^2 + 2 +1\right) $
$\qquad =2^{n-1}\times 2 - \frac{2^{n-1}-1}{2-1} \because T(1) = 2, S_n = \frac{a.\left(r^n -1\right)}{r-1}$
$\qquad =2^n - \left(2^{n-1} -1\right)$
$\qquad =2^{n-1} + 1$