Mathematical Induction – JEE Main Previous Year Question with Solutions
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Q. Statement – 1: For each natural number $$n,(n+1)^{7}-n^{7}-1 \text { is divisible by } 7 \text { . }$$Statement – 2: For each natural number $n, n^{7}-n$ is divisible by 7(1) Statement-1 is false, statement-2 is true.(2) Statement-1 is true, statement-2 is true; Statement-2 is correct explanation for statement-1.(3) Statement-1 is true, statement-2 is true; Statement-2 is not a correct explanation for statement-1.(4) Statement-1 is true, statement-2 is false. [AIEEE-2011]

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Sol. (2)Statement-2:$P(n)=n^{7}-n$Put $n=1, \quad 1-1=0$ is divisible by 7Let $n=k, \quad P(k)=k^{7}-k$ is divisible by 7Put $n=k+1$$\mathrm{P}(\mathrm{k}+1)=(\mathrm{k}+1)^{7}-(\mathrm{k}+1)$$=\mathrm{k}^{7}+^{7} \mathrm{C}_{1} \mathrm{k}^{6}+\ldots \ldots+^{7} \mathrm{C}_{6} \mathrm{k}+1-\mathrm{k}-1$$=\left(\mathrm{k}^{7}-\mathrm{k}\right)+ multiple of 7\Rightarrow \mathrm{P}(\mathrm{k}+1) is divisible by 7Hence \mathrm{P}(\mathrm{n})=\mathrm{n}^{7}-\mathrm{n} is divisible by 7Now statement- 1\mathrm{P}(\mathrm{n})=(\mathrm{n}+1)^{7}-\mathrm{n}^{7}-1$$(\mathrm{n}+1)^{7}=(\mathrm{R}+1)^{7}-\left(\mathrm{n}^{7}-1\right.$$\quad$ divisible by $7 \quad$ is divisible by 7$\Rightarrow \mathrm{P}(\mathrm{n})=(\mathrm{n}+1)^{7}-\mathrm{n}^{7}-1$ is divisible by 7

• January 8, 2021 at 10:53 pm

Only one question.what ???????

• April 18, 2021 at 8:48 pm

Very easy question. Only 15 sec required to solve. Post some excellently difficult question please

• July 13, 2020 at 8:15 pm

Is any other questions came from this topic?

• March 3, 2021 at 9:48 pm

Good for jee preparation

• May 31, 2020 at 10:28 pm

Poor

• May 21, 2020 at 1:24 pm

Excellent

• May 10, 2020 at 4:38 pm