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Using the principle of mathematical induction, prove that `:``1. 2. 3+2. 3. 4++n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4^`for all `n in N`. |
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Answer» Let `P(n):(1)/(1.2.3)+(1)/(2.3.4)+.....+(1)/(n(n+1)(n+2))=(n(n+3))/(4(n+1)(n+2))` .....(i) Step I For n=1. LHS is Eq. (i) `=(1)/(1.2.3)=(1)/(6)` and RHS of Eq. (i) `=(1(1+3))/(4(1+1)(1+2))=(1)/(6)` Therefore , P(1) is true . Step II Assume that P(k) is ture , then `P(k):(1)/(1.2.3)+(1)/(2.3.4)+......+(1)/(k(k+1)(k+2))=(k(k+3))/(4(k+1)(k+2))` Step III For `n=k+1`, ``P(k):(1)/(1.2.3)+(1)/(2.3.4)+.....+(1)/(k(k+1)(k+2))+(1)/((k+1)(k+2(k+3)))=((k+1)(k+4))/(4(k+2)(k+3))` `therefore LHS =(1)/(1.2.3)+(1)/(2.3.4)+.....+(1)/(k(k+1)(k+2))+(1)/((k+1)(k+2)(k+3))` ` =(k(k+3))/(4(k+1)(k+2))+(1)/((k+1)(k+2)(k+3))` [by assumption step ] `=(k(k+3)^2+4)/(4(k+1)(k+2)(k+3))` `=(k^3+6k^2+9k+4)/(4(k+1)(k+2)(k+3))` `=((k+10^2(k+4))/(4(k+1)(k+2)(k+3))` `=((k+1)(k+4))/(4(k+2)(k+3))=RHS` Therefore , `P(k+1)` is true , Hence , by the principle of mathematical P(n) is true for all `n in N`. |
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