If ( 1 + x + x 2   ) 25   =   a 0 + a 1   x + a 2   x 2 +   . . . . . . . . . . . . . a 50   x 50 , then the value of a 0 + a 2 + a 4 +   . . . . . . . . . + a 50   is

Step-by-step solution

Putting x = 1 , we get, 3 25 = a 0 + a 1 + a 2 + ..... + a 50 ....(1) Putting x = − 1 , we get, 1 = a 0 − a 1 + a 2 + ........ + a 50 ...(2) adding equation (1) & (2), we get, ⇒ a 0 + a 2 + … … + a 50 = 3 25 + 1 2 = ( 1 + 2 ) 25 + 1 2 ⇒ a 0 + a 2 + a 4 + ..... + a 50 = ​ ( 25 C 0 + ​ 25 C 1 · 2 +   ​ 25 C 2 · 2 2 + … +   ​ 25 C 25 · 2 25 ) + 1 2 = 2 [ 1 + ​ 25 C 1 +   ​ 25 C 2 · 2 + … +   &#82