Question

# With the help of mathematical induction find for all n≥1 the sum of the series 11.2+12.3...1n.n+1 is equal ton+1nn+12nn+1n3n+1

Solution

## The correct option is B nn+1For any integer n≥1 , let pn be the statement that 11.2+12.3...1n+1=nn+1 . Base case––––––––––– : The statement P1 says that 11.2=11+1, This is true. Inductive step––––––––––––––––: Fix k≥1 , and suppose that Pk holds, that is, 11.2+12.3...1k(k+1)=kk+1 . It remains to show that Pk+1 holds, that is, 11.2+12.3...1(k+1)(k+2)=k+1k+2 . Since we know the sum of the series till the kth term we can write the LHS as  kk+1+1(k+1)(k+2) =k2+2k+1(k+1)(k+2) =(k+1)2(k+1)(k+2) =k+1k+2 = RHS Therefore Pk+1 holds Thus, by the principle of mathematical induction, for all n≥1, Pn holds.

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