T(n) = 2T($\frac{n}{2}$)+n ; n>1
= 1 ; . n=1
put $\frac{n}{2}$ in place of n ===> T( $\frac{n}{2}$ ) = 2 T($\frac{n}{4}$) + $\frac{n}{2}$ ===> T( $\frac{n}{2}$ ) = 2 T($\frac{n}{2^{2}}$) + $\frac{n}{2}$ ---- (1)
SUBSTITUTE (1) in the original equation
T(n) = 2 ( 2 T($\frac{n}{2^{2}}$) + $\frac{n}{2}$ ) + n
T(n) = 22 T($\frac{n}{2^{2}}$) + 2 . $\frac{n}{2}$ + n
T(n) = 22 T($\frac{n}{2^{2}}$) + n + n = 22 T($\frac{n}{2^{2}}$) + 2 n
T(n) = 23 T($\frac{n}{2^{3}}$) + 3 n
........
T(n) = 2k T($\frac{n}{2^{k}}$) + k n ------------- (2)
for matching base condition T($\frac{n}{2^{k}}$) = T(1)
$\frac{n}{2^{k}}$ = 1 ====> n= 2k ====> k = log2 n -----> substitute this in (2)
T(n) = n T( 1 ) + ( log2 n ) n = n + n log2 n =O ( n log2 n )