CBSE Class 9 Answered

         Since ( 6(1+2)+ 7(2+1)) = 63  + 73  = 559 = 43 X 13, 
the formula is true for n = 1.
		

                                             
Assume that P(n) is true for n = k, that is 6(k+1)+ 7(2k+1)) = 43x 
for some integer x.
		

Now show that the formula is true for n = k + 1. 

Observe that 
          
P(k+1) = 6(k+1+2)+ 7(2(k+1)+1)

          = 6(k+3)+ 7(2k+3)

              
       = 6 X 6(k+2)+ 72  X 7(2k+1)

              
       = 6 X 6(k+2)+(6+43) X 7(2k+1)

              
       = 6 X 6(k+2)+ 6 X 7(2k+1)+ 43 X 7(2k+1)

                
       = 6 X ( 6(k+2)+ 7(2k+1) )+ 43 X 7(2k+1)

                              
       = 6 X ( P(k) ) + 43 X 7(2k+1)