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Question

With the help of mathematical induction find for all n1 the sum of the series 11.2+12.3...1n.n+1 is equal to
  1. n+1
  2. nn+1
  3. 2nn+1
  4. n3n+1


Solution

The correct option is B nn+1

For any integer n1 , 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 k1 , 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 n1, Pn holds.

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