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If `a_(1)=1,a_(n+1)=(1)/(n+1)a_(n),a ge1`, then prove by induction that `a_(n+1)=(1)/((n+1)!)n in N`. |
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Answer» Let `P(n):a_(n+1)=(1)/((n+1)!),n in N`.....(i) where `a_1=1 and a_(n+1)=(1)/((n+1))a_(n),n ge 1` ......(ii) Step I For n=1, form Eq. (i) , we get `a_(2)=(1)/((1+1)!)=(1)/(2!)` But from Eq. (ii) , we get a_(2)=(1)/((1+1)),a_(1)=(1)/(2)(1)=(1)/(2)` which is true , Also, for n=2 from Eq. (i) we get `a_3=(1)/(3!)=(1)/(6)` But from Eq. (ii) , we get `a_3=(1)/(3),a_2=(1)/(3).(1)/(2)=(1)/(6)` which is also true . Hence ,P(1) and |
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