First slide
Introduction to P.M.I

The sum of series 13+232+333++n3n is


Let the given statement be P(n). 


Step l: Let it is true for n = 1, 


    3=1.3  which is true

Step II: For n=k, 

 i.e. 13+232+333++k3k=(2k1)3k+1+34-----i

Step III: For n=k+1,

13+232+333++k3k+(k+1)3k+1 =(2k1)3k+1+34+(k+1)3k+1[ using Eq. (i)]  =(2k1)3k+1+3+4(k+1)3k+14

On taking 3k+1 common in first and last term of numerator part, 


On taking 3 common in first term of numerator part,


Therefore, P(k + 1) is true when P(k) is true. 

Hence, from the principle of mathematical induction, the statement is true for all natural numbers n.

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