• # question_answer Statement -1 : For each natural number $n,{{(n+1)}^{7}}-{{n}^{7}}-1$is divisible by 7. Statement - 2 : For each natural number $n,{{n}^{7}}-n$is divisible by 7.     AIEEE  Solved  Paper (Held On 11 May  2011) A)  Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1. B)  Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1 C)  Statement-1 is true, Statement-2 is false D)  Statement-1 is false, Statement-2 is true

Statement 2 $P(n)={{n}^{7}}-n$ Put n = 1        1 - 1 = 0 is divisible by 7 Let n = k        $P(k)={{k}^{7}}-k$is divisible by 7 Put $n=k+1$ $\therefore$$P(k+1)={{(k+1)}^{7}}-(k+1)$ $={{k}^{7}}{{+}^{7}}{{C}_{1}}{{k}^{6}}{{+}^{7}}{{C}_{2}}{{k}^{5}}+.....{{+}^{7}}{{C}_{6}}k+1-k-1$ $P(k+1)=({{k}^{7}}-k)+$multiple of 7 As 7 is coprime with 1,2,3,4,5,6 so $^{7}{{C}_{1}}{{,}^{7}}{{C}_{2}},{{.....}^{7}}{{C}_{6}}$are all divisible by 7 Hence $P(n)={{n}^{7}}-n$is divisible by 7 Statement 1 ${{n}^{7}}-n$is divisible by 7 $\Rightarrow$${{(n+1)}^{7}}-(n+1)$ is divisible by 7 $\Rightarrow$${{(n+1)}^{7}}-{{n}^{7}}-1+({{n}^{7}}-n)$ is divisible by 7 $\Rightarrow$${{(n+1)}^{7}}-{{n}^{7}}-1$is divisible by 7