InterviewSolution
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InterviewSolution
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`1^(2)+3^(2)+5^(2)+.......+(2n-1)^(2)` `=(n(2n-1)(2n+1))/(3)` |
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Answer» `"Let "P (n) : 1^(2)+3^(2)+5^(2)+......+(2n-1)^(2)` ` =1/3 n(4n^(2)-1)` for n =1 `L.H.S. =1^(2) =1, R.H.S. =1/3 .1.(4.1^(2)-1)=1` `:. L.H.S. =R.H.S.` `rArr` P(n) is true for n=1 Let P(n) be true for n=K `:. P(k) : 1^(2) +3^(2)+5^(2)+` `.......+(2k-1)^(2)=1/3 k(4k^(2)-1)` for n=K+1 `P(k+1) : 1^(2) +3^(2) +5^(2)+......+(2K-1)^(2)` `+(2K+1)^(2)=1/3 k(4k^(2)-1)+(2k+1)^(2)` `(k(4k^(2)-1)+3(2k+1)^(2))/(3)` `=1/3[4k^(3)-k+12k^(2)+12k+3]` `=1/3[4k^(3)+12k^(2)+11k+3]` `=1/3(k+1)(4k^(2)+8k+3)` `=1/3(k_+1)[4(k^(2)+2k+1)-1]` `=1/3(k+1)[4(k+1)^(2)-1]` `rArr` P (n) is also true for n=K+1 Hence from the principle of mathematical induction P (n) is true for all natural numbers. n |
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