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`1.2.3+2.3.4+. . .+n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/(4)` |
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Answer» Let `P(n):1.2.3+2.3.4+.....+n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/(4)` ........(i) Step I For `n =1, LHS of Eq. (i) `=1.2.3=6`and RHS of Eq. (i) `=(1.(1+1(1+2)(1+3))/(4)=6` `therefore` LHS = RHS Therefore , P(1) is true , then Step II Assume that P(k) is true , then `P(k):1.2.3+2.3.4+.....+k(k+1)(k+2)=(k(k+1)(k+2)(k+3))/(4)` Step III For `n=k+1` `P(k+1):1.2.3+2.3.4+.....+k(k+1)(k+2)+(k+1)(k+2)(k+3)` `=((k+1)(k+2)(k+3)(k+4))/(4)` `therefore LHS = 1.2.3+2.3.4+...+k(k+1)(k+2)+(k+1)(k+2)(k+3)` =(k(k+1)(k+2)(k+3))/(4)+(k+1)(k+2)(k+3)` [by assumption step] `=((k+1)(k+2)(k+3))/(4)(k+4)` `=((k+1)(k+2)(k+3)(k+4))/(4)=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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