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
Correct Answer: A
Solution :
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
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