$T(n) = T(n-1) + 2^{n}$

$T(n) = T(n-2) + 2^{n-1} + 2^{n}$

$T(n) = T(n-3) + 2^{n-2} + 2^{n-1} + 2^{n}$

$T(n) = T(n-k) + 2^{n-k+1}+.... + 2^{n-1} + 2^{n}$

$\text{Now at }k = n - 1$,

$T(n) = T(1) + 2^{2}+.... + 2^{n-1} + 2^{n}$

$T(n) = 1 + 2^{2}+.... + 2^{n-1} + 2^{n}$

$T(n) = \left ( 2^{0} + 2^{1} + 2^{2}+.... + 2^{n-1} + 2^{n} \right ) - 2^{1}$

$T(n) = 2^{n+1} - 1 - 2$

$T(n) = 2^{n+1} - 3$

$\therefore T(10) = 2^{10+1} - 3 = 2048 - 3 = 2045 $