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)=9 k where k N , then find k .

Question

Quantrex Academy · Accepted Answer

Correct answer: 2187

We know that (a-1)^n= ^n C₀ a^n- ^n C₁ a^ n-1 + ^n C₂ a^ n-2 - . .+(-1)^ n-1 ^n C_ n-1 a+(-1)^n C_n (a-1)^n a = ^n C₀ a^ n-1 - ^n C₁ a^ n-2 + ^n C₂ a^ n-3 - +(-1)^ n-1 ^n C_ n-1 + (-1)^n a ^n C_n Hence f(n)= (a-1)^n-(-1)^n a aligned Now, f(2007)+f(2008)= (a-1)²⁰⁰⁷+1 a + (a-1)²⁰⁰⁸-1 a & = (a-1)²⁰⁰⁷(1+a-1) a =(a-1)²⁰⁰⁷ & = (3^ 1 223 )²⁰⁰⁷=3^9=3^2 3^7=9(2187) aligned Hence k =2187 Ans.

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)=9 k where k N , then find k .

Correct answer

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