If (1^4+ 1 4 ) (3^4+ 1 4 ) (2 n-1)^4+ 1 4 (2^4+ 1 4 ) (4^4+ 1 4 ) (2 n)^4+ 1 4 = 1 k₁ n^2+k₂ n+k₃ , then k₁-k₂+k₃ is equal to

Step-by-step solution

aligned & m^4+ 1 4 = (m^2+m+ 1 2 ) (m^2-m+ 1 2 ) & _ k=1 ^n= (2 k-1)^4+ 1 4 (2 k)^4+ 1 4 & = _ k=1 ^n= ((2 k-1)^2+(2 k-1)+ 1 2 ) ((2 k-1)^2-(2 k-1)+ 1 2 ) ((2 k)^2+2 k+ 1 2 ) ((2 k)^2+2 k+ 1 2 ) & = 1 8 n^2+4 n+1 & k₁=8 . k₂=4 . k₃=1 . k₁-k₂+k₃=8-++1=5 aligned