If S₁, S₂, S₃ ........S_p are the sums of n terms of p A.P.’s whose first terms are 1, 2,3, .....p and common differences are 1, 3, 5, .....(2p – 1) respectively. Show that

         

Here,                 = Sum of first n terms  of an A.P. with first term 1 and c.d. 1.
                             =
                         Sum of first n terms of an A.P. with first term 2 and c.d. 3.
                             =
                       =  Sum of first n terms of an A.P. with first term 3 and c.d. 5.
                             =
                            .............................................
                           ..............................................
                      Sum of first n terms of an A.P. with first term p and c.d. (2p - 1)
                          
∴     = 
                                              =    
                                                   
                                               = 
                                               =