InterviewSolution
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InterviewSolution
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Use the principle of mathematical induction to prove that for all `n in N` `sqrt(2+sqrt(2+sqrt2+...+...+sqrt2))=2cos ((pi)/(2^(n+1)))` When the LHS contains n radical signs. |
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Answer» Let P(n) =`sqrt(2+sqrt(2+sqrt2+...+...+sqrt2))=2cos ((pi)/(2^(n+1))).....(i)` Step-1 For n=1 `"LHS of "(i)=sqrt2 and "RHS of (i) "=2cos((pi)/(2^(2)))` `2cos((pi)/(4))` `2.(1)/(sqrt2)` `=sqrt2` Therefore, P(1) is true. Step II. Assume it is true for n=k, `P(k)=underset("k radical sign")(sqrt(2+sqrt(2+sqrt2+...+...+sqrt2)))=2cos ((pi)/(2^(n+1)))` `=sqrt({2+P(k)})` `sqrt(2+2cos((pi)/(2^(k+1))))" "("By assumption step")` `sqrt(2(1+cos((pi)/(2^(k+1)))-1))` `sqrt(4cos((pi)/(2^(k+2))))` `2cos((pi)/(2^(k+2)))` This shows that the result is true for n=k+1. Hence by the principle of mathematical, induction, the result is true for all `n in N`. |
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