How can we prove that :

(1 + 1 / 3) (1 + 5 / 4)(1 + 7 / 9)........{(1 + (2n + 1 / n2)} = (n + 1)2 for all n Є N.

Please answer this question!!!!!!!!!!

hi dude iam arjun of class 11 from chennai,seee

first (n=1)

i.e, 1+(2.1+1)/12 = (1+1)2

      1+3                 =22

       4             =          4 so p(n)=1 is true

now n=k

             (1 + 1 / 3) (1 + 5 / 4)(1 + 7 / 9) ...... 1+(2.k+1)/k= (k+1)2       keep this as - 1

 

now n=k+1

               1+(2(k+1)+1)/(k+1)2 = (k+2)2

 

=(1 + 1 / 3) (1 + 5 / 4)(1 + 7 / 9)……1+(2(k+1)+1)/(k+1)2 = (k+2)2

 (1 + 1 / 3) (1 + 5 / 4)(1 + 7 / 9)……(1+(2k+3)/(k+1)2) = (k+2)2

 (1 + 1 / 3) (1 + 5 / 4)(1 + 7 / 9)……(1+(2k+1)/(k)2) (1+(2k+3)/(k+1)2) = (k+2)2 --

using 1

 

 

L.H.S   R.H.S

(k+1)2 (1+(2k+3)/(k+1)2)  =  (k+2)2

(k+1)2  (k+1)2  + (2k+3)

         ----------

  (K+1)2 

So, (k+1)2  + (2k+3)

=  k2+2k+12  + 2k+3

=  k2+4k+4  (splitting the middle term)

=  (k+2) (k+2)

=  (k+2)2

*  L.H.S = (k+2)2  &   R.H.S = (k+2)2

L.H.S = R.H.S i.e p(k+1) is true for P(K)  J :D

 

             

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