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`1^(3)+2^(2)+3^(3)+. . .+n^(3)=((n(n+1))/(2))^(2)`. |
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Answer» Let `P(n):1^3+2^3+3^3+.....+n^3=[(n(n+1))/(2)]^2`......(i) Step I For `n=1`, LHS of Eq.(i) `=^3=1` and RHS of Eq. (i). `[(1(1+1))/(2)]^2=1^2=1` `therefore LHS=RHS` Therefore ,P(1), is ture. Step II Assume P(k) is true , then `P(k):1^3+2^3+3^3+.....K^3=[(k(k+1))/(2)]^2` Step III For `n=k+1`, `P(k+1):1^3+2^3+3^3+......+^3(k+1)^3` `=[((k+1)+(k+2))/(2)] ^2` LHS `=1^3+2^3+3^3+.....+k^3+(k+1)^3=[(k(k+1))/(2)]^2+(k+1)^3` [by assumption step] `=((k+1)^2)/(4)[k^2+4(k+1)]` `=((k+1)^2(k^2+4k+4))/(4)` `=((k+1)^2(k+2)^2)/(4)` `=[(k+1(k+2))/(2)]^2=RHS` Therefore , `P(k+1)` is true , Hence , by the principle of mathematical induction , P(n)is true for all `n epsi N`. |
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