Let a_k = ( _k) i + j and b_k = i - ( _k) j , where _k = 2^ k-1 2^n + 1 , for some n N , n > 5 . Then the value of _ k=1 ^ n | a_k |^2 _ k=1 ^ n | b_k |^2 is _____.

Step-by-step solution

Let a _k = _k , i + j and b _k = i - _k , j , where _k = 2^ k-1 2^n + 1 . | a _k|^2 = ^2 _k + 1 = ^2 _k | b _k|^2 = 1 + ^2 _k = ^2 _k So the required ratio is: _ k=1 ^n | a _k|^2 _ k=1 ^n | b _k|^2 = _ k=1 ^n ^2 _k _ k=1 ^n ^2 _k Relation between successive angles: _ k+1 = 2^k 2^n + 1 = 2 _k Key identity: ^2 + ^2 = 1 ^2 + 1 ^2 = 1 ^2 ^2 = 4 ^2 2 = 4 ^2 2 Hence: ^2 = 4 ^2 2 - ^2 Evaluating the numerator using this identity and 2 _k = _ k+1 : _ k=1 ^n ^2 _k = _ k=1 ^n (4 ^2 _ k+1 - ^2 _k ) = 4 _ k=1 ^n ^2 _ k+1 - _ k