Ch. 6. Sequences and Series - 2

Arithmetico Geometric sequence

Arithmetico-geometric sequence

When a1, a2, a3,…an is an A.P. and b1,b2,b3,…,bn are in GP
a1b1, a2b2, a3b3 Are said to be in arithmetico-geometric sequence.

In terms common difference and common ratio, the series can be written as

ab, (a+d)br, (a+2d)br²

nth term = [a+(n-1)d]br^n-1

Sum to n term of an A-G series

Sn = ab+ (a+d)br+ (a+2d)br²+…+[a+(n-2)d]br^n-2 + [a+(n-1)d]br^n-1

Multiply by each of Sn by r
r*Sn = abr + (a+d)br²+ (a+2d)br³+…+[a+(n-2)d]br^n-1
+ [a+(n-1)d]brⁿ


Sn – r*Sn = ab+dbr+d br² +d br³ +…+d br^n-1 - [a+(n-1)d]brⁿ

=> ab+dbr[1+r +r² +r³ +…+r^n-2] - [a+(n-1)d]brⁿ

As [1+r +r² +r³ +…+r^n-2] = (1-r^n-1)/(1-r) (as there are n-1 terms in the GP)

(1-r)Sn = ab + dbr(1-r^n-1)/(1-r) - [a+(n-1)d]brⁿ

Sn = ab/(1-r) + dbr(1-r^n-1)/(1-r)² - [a+(n-1)d]brⁿ/(1-r)

Sum to n terms of an A-G series =

Sn = ab/(1-r) + dbr(1-r^n-1)/(1-r)² - [a+(n-1)d]brⁿ/(1-r)