First divide the left side equally.

Reciprocity method

The recurrence formula is in the form of quotient: an+1= (pan+b)/(qan+c) (an ≠ 0, PQ ≠ 0, PC ≠ QB).

If b=0, an+ 1 = pan/(qan+c). Because an≠0, the reciprocal of both sides is 1/an+ 1=q/p+c/pan, so bn =1/an.

If b≠0, let an+ 1+x=y(an+x)/qan+c, compare with the known recurrence formula to get x and y, let bn=an+x, and get bn+ 1=ybn/qan+c, which is converted into b=0.

Type 3

structured approach

The recurrence formula is pan=qan- 1+f(n)(p and q are nonzero constants), and a new geometric series solution can be constructed by using the undetermined coefficient method.