For example the sequence 1+3x+6x2+9x3+12x4+15x5+18x6 is an arithmetico geometric sequence where the sequence 1 + 3 + 6 + 9 + 12 + 15 + 18 is in arithmetic progression and the sequence 1+x+x2+x3+x4+x5+x6 is in geometric progression.
Sum of n terms of an Arithmetico Geometric Series:
Sn=a+[a+d]r+[a+2d]r2+[a+3d]r3+[a+4d]r4+.......+[a+(n–1)d]rn–1 [ Where d ≠ 0 and r ≠ 0 ] . . . . . . (1)
Now, multiplying the above equation by ‘r’ we get
r.Sn=ar+[a+d]r2+[a+2d]r3+[a+3d]r4+[a+4d]r5+.......+[a+(n–1)d]rn [ Where d ≠ 0 and r ≠ 0 ] . . . . . . (2)
Now, Equation (2) – Equation (1) we get
Or, Sn (1 – r) = a + d[1−rr(1−rn−1)] – [a + (n – 1) d] × r n
Sum of an Infinite Arithmetico Geometric Series:
If n→∞and∣r∣<1 then, rn = 0.
The Method of Differences:
Suppose p1,p2,p3,p4,.....pn is a given sequence such that p2–p1,p2–p3,.....,pn–pn−1 is either in an arithmetic or geometric progression, then, the sum of the given sequence can be evaluated by following the steps mentioned below: