If f(x)= 2^x 2^x+ 2 , x R , then _ k =1 ⁸¹ f ( k 82 ) is equal to

Step-by-step solution

aligned & f(x)= 2^x 2^x+ 2 & f(x)+f(1-x)= 2^x 2^x+ 2 + 2^ 1-x 2^ 1-x + 2 & = 2^x 2^x+ 2 + 2 2+ 2 2^x = 2^x+ 2 2^x+ 2 =1 aligned aligned & Now, _ k =1 ⁸¹ f ( k 82 )= f ( 1 82 )+ f ( 2 82 )+ + f ( 81 82 ) & = f ( 1 82 )+ f ( 1 82 )+ + f (1- 2 82 )+ f (1- 1 82 ) aligned aligned & [ f ( 1 82 )+ f (1- 1 82 ) ]+ [ f ( 2 82 )+ f (1- 2 82 ) ]+ .40 cases + f ( 41 82 ) & (1+1+ .+1) 40 times + 2^ 1 / 2 2^ 1 / 2 +2^ 1 / 2 & 40+ 1 2 = 81 2 aligned