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

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Comments
  • January 8, 2021 at 10:53 pm

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

    0
    • April 18, 2021 at 8:48 pm

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

      0
  • July 13, 2020 at 8:15 pm

    Is any other questions came from this topic?

    1
    • March 3, 2021 at 9:48 pm

      Good for jee preparation

      0
  • May 31, 2020 at 10:28 pm

    Poor

    0
  • May 21, 2020 at 1:24 pm

    Excellent

    0
  • May 10, 2020 at 4:38 pm

    Very bad

    0