Let a=3^ 1 / 223 +1 and for all n 3 let f(n)= ^n c₀ a^ n-1 - ^n c₁ a^ n-2 + ^n c₂ a^ n-3 - . .+(-1)^ n-1 ^n c_ n-1 a^0 if the value of f(2007)+f(2008)=3^k where k N then K is

Step-by-step solution

aligned & f(n)= ^n c₀ a^ n-1 - ^n c₁ a^ n-2 + ^n c₂ a^ n-3 - . .+(-1)^ n-1 ^n c_ n-1 a^a & = 1 a ( ^n c₁ a^n- ^n c₁ a^ n-1 + ^n c₂ a^n+ +(-1)^n c_ n-1 a ) & = 1 a ((a-1)^n-(-1)^n ^n c_n )= 1 a ( 1 3^ n 23 -(-1)^n ) & f(x)= 3^ x / 223 -(-1)^n (3^ 1 / 223 +1 ) & f(2007)= (3^ 2007 / 223 +1 ) (3^ 1 / 223 +1 ) & f(2008)= (3^ 2008 / 223 -1 ) (3^ 1 / 223 +1 ) & f(2007)+f(2008)=3^9 then k=9 aligned