Let S_n denotes the sum of n terms of an A.P. whose first  term is a. If the common difference 

d = S_n – k S_{n – 1} +  S_{n – 2} then k =

We have , ${\mathrm{a}}_{\mathrm{n}}={\mathrm{S}}_{\mathrm{n}}-{\mathrm{S}}_{\mathrm{n}-1}$     (1)

and  ${\mathrm{a}}_{\mathrm{n}-1}={\mathrm{S}}_{\mathrm{n}-1}-{\mathrm{S}}_{\mathrm{n}-2}$       (2)

$\begin{array}{l}\therefore \mathrm{d}={\mathrm{a}}_{\mathrm{n}}-{\mathrm{a}}_{\mathrm{n}-1}\\ =\left({\mathrm{S}}_{\mathrm{n}}-{\mathrm{S}}_{\mathrm{n}-1}\right)-\left({\mathrm{S}}_{\mathrm{n}-1}-{\mathrm{S}}_{\mathrm{n}-2}\right)\\ \left[\text{ From (1) and (2) }\right]\\ ={\mathrm{S}}_{\mathrm{n}}-2{\mathrm{S}}_{\mathrm{n}-1}+{\mathrm{S}}_{\mathrm{n}-2}.\end{array}$