The sum of first n terms of a certain series is given as 2n2-3n . Show that the series is an A.P.

The sum of first n terms of a certain series is given as 2n2-3n . Show that the series is an A.P.

Solution :

Given that :

tn  = 2n2 - 3n

tn  = 2n2 - 3n n = 1 t1  = 2(1)2 - 3(1)   =   2 - 3    t1   =  -1 =  a

n = 2 t2  = 2(2)2 - 3(2)   =   8 - 6    t2   =  2 d  = 2 - (-1)  = 3

Sn  = (n/2) [2a + (n - 1)d]

Sn  = (n/2) [2(-1) + (n - 1)3]

  =  (n/2) [-2 + 3n - 3]

  =  (n/2) [3n - 5]