InterviewSolution
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InterviewSolution
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`1+1/(1+2)+1/(1+2+3)+1/(1+2+3+n)=(2n)/(n+1)` |
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Answer» Let P (n) `:1 +(1)/(1+2)+(1)/(1+2+3)+……` `+(1)/((1+2+3+.....+n))=(2n)/(n+1)` ltbr. For n=1 `L.H.S. =1` `R.H.S. =(2.1)/(1+1) =(2)/(2) =1` `:. " "L.H.S.=R.H.S.` Therefore,P (n) is true for n=1 Let P (n) true for n =K . `P(k) :1 (1)/(1+2) +(1)/(1+2+3)+....` `+(1)/(1+2+3+....+K)=(2K)/(K+1)` For n =k +1 `P (k+1) =1 + (1)/(1+2)+(1)/(1+2+3)+.....+(1)/(1+2+3+.....+K)` `+(1)/(1+2+3+.....+K+(K+1))` `(2K)/(K+1)+(1)/(1+2+3+......+K+(K+1))` `=(2k)/(K+1) +(1)/((K+1)(K+2))=(2K(K+2)+2)/((K+1)(K+2))` `=(2(K^(2)+2K+1))/((k+1)(K+2)) =(2(K+1)^(2))/((K+1)(K+2))` `(2(K+1))/(K+2)=(2(K+1))/((K+1)+1)` `rArr` P (n) is also true for n=k+1 hence form the principle of mathematical induction P (n) is true for all natural numbers n. |
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