InterviewSolution
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InterviewSolution
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Shwo that `n^3+(n+1)^3+(n+2)^3` is divisible 9 for everynatural number n. |
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Answer» Let f(n)=`n^(3)+(n+1)^(3)+(n+2)^(3)` Let P(n): `f(n)i.e. n^(3)+(n+1)^(3)+(n+2)^(3)` is divisible by 9. Now, f(1)=`1^(3)+2^(3)+3^(2)=36` is divisible by 9. `therefore P(1) ` is true …….(A) Let P(m) be true `Rightarrow f(m)=m^(3)+(m+1)^(3)+(m+2)^(3)` is divisible by 9. `Rightarrow f(m)=m^(3)+(m+1)^(3)+(m+2)^(3)=9k` where k is an integer ...(i) To prove P(m+1) is true i.e. f(m+1) is divisible by 9. Now, `f(m+1)=m(m+1)^(3)+(m+2)^(3)+(m+3)^(3)` `=(m+1)^(3)+(m+2)^(3)+m^(3)+9m^(2)+27m+27` `=[m^(3)+(m+1)^(3)+(m+2)^(3)]+9m^(2)+27m+27` `=9k+9(m^(2)+3m+3)` which is divisible by 9 Hence P(m+1) is true whenever P(m) is true. .........(B) From (A) and (B). It follows that P(n) is true for all natural number n. |
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