InterviewSolution
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InterviewSolution
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Prove that `n(n+1)(n+5)` is a multiple of 3. |
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Answer» Let P(n) =n(n+1) (n+5) For n=1 P(1)=1.(1+1)(1+5)=12=3(4) Which is a multiple of 3. `rArr` p (n) is true for n=1 LetP (n) be true for n=K `:. P(k) =k(k+1)(K+05)=3lambda "(say) "` `" Where " lambda in I` For n=K+1 `P(k+1) =(k+1)(K=2)(K+6)` `=(K+1)[K^(2)+8K+12]` `=(k=1)[K^(2)=5k)+(3k+12)]` `=(k+1)k(k+5)+3(k+1)(+4)` `=3lamda+3(K+1)(K+4)` [From equation (1)] `=3[lambda+(k=1)(K+4)]` Which is a multiple of 3. `rArr` P(n) is also true for n=K+1 Hence by the principle of mathematical induction P (n) is true for all natural numbers n. |
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