InterviewSolution
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InterviewSolution
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Prove that `1+2+3+4........+N |
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Answer» `" Let " P(n) : 1+2+3 +....+ nlt1/8(2n+1)^(2)` for n=1 `L.H.S. =1, R.H.S. =1/8 (2+1)^(2)=9/8` `:. L.H.S. lt R.H.S. ` `rArr` P(n) is true for n=1 Let P(n) be true for n=K `P(k) : 1 +2+3+….+klt 1/8 (2k+1)^(2) ……(1)` `n=K+1` `P(k+1) : 1+2+3 +.....+K+(K+1)` ` lt 1/8 (2k+1)^(2)+(K+1) " "["From eauation "(1)]` `=((2k+1)^(2)+8(k+1))/(8)` `=(4k^(2)+4k+1+8K+8)/(8) =(4k^(2)+12K+9)/(8)` `=1/8 (2k+2)^(2) =1/8 {2(k+1)+1}^(2)` `rArr` P(n) is also true for n=K+1 Hence from the principle of mathematical induction the given statement is true for all `n in N` |
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