# Arithmetico-Geometric Sequence

Let

*a*, (*a*+*d*)*r*, (*a*+ 2*d*)*r*^{2}, (*a*+ 3*d*)*r*^{3}, â€¦, be an arithmetic- geometric sequence.Then,

*a*+ (*a*+*d*)*r*+ (*a*+ 2*d*)*r*^{2}+ (*a*+ 3*d*)*r*^{3}+ â€¦ is an arithmetico-geometric series.**Sum of**

*n*terms of an arithmetico-geometric sequence(1)

If the students find it difficult to apply this formula, then it is advised to find the sum by the following mechanism:

Write the series as sum

*S*=*a*+ (*a*+*d*)*r*+ (*a*+ 2*d*)*r*^{2}+ (*a*+ 3*d*)*r*^{3}+ â€¦Now write the series after multiplying by common ratio

*r*, as*rS*=*ar*+ (*a*+*d*)*r*^{2}+ (*a*+ 2*d*)*r*^{3}+ (*a*+ 3*d*)r^{4}+ â€¦Now subtract the

*rS*from*S*and find the sum of resulting GP.**Sum of an infinite arithmetico-geometric sequence**

If |

*r*| < 1, then*r*,^{n}*r*^{n}^{â€“1}â†’ 0 as*n*â†’ âˆž and it can also be shown that*n*â‹…*r*â†’ 0 as^{n}*n*â†’ âˆž. So, from (1), we obtain that*S*â†’ , as_{n}*n*â†’ âˆž.# Method of difference

Consider the sequence

*S*:*a*_{1},*a*_{2},*a*_{3},*a*_{4}, â€¦.If difference series

then general term of

We can find values of

*a*_{2}â€“*a*_{1},*a*_{3}â€“*a*_{2},*a*_{4}â€“*a*_{3}, â€¦ is AP,then general term of

*S*is*a*=_{n}*An*^{2}+*Bn*+*C*, where*A*,*B*,*C*are constant.We can find values of

*A*,*B*,*C*by putting*n*= 1, 2, 3.If difference series

We can find values of

*a*_{2}â€“*a*_{1},*a*_{3}â€“*a*_{2},*a*_{4}â€“*a*_{3}, â€¦ is GP with common ratio*r*, then general term of*S*is*a*=_{n}*Ar*+^{n}*Bn*+*C*, where*A*,*B*,*C*are constants.We can find values of

*A*,*B*,*C*by putting*n*= 1, 2, 3.

** Notes: **For any series if

*S*, sum of

_{n}*n*terms, is given then we can find the

*n*th term from

*t*=

_{n}*S*â€“

_{n}*S*

_{n}_{ â€“ 1}.