I think you'll find this useful:
PlanetMath: arithmetic-geometric series
The general arithmetic-geometric series of n terms is a series of the form:
a + (a+d)r + (a + 2d)r^2 + ... +[a + (n - 1)d]r^n-1 Where the first factors of each term form an arithmetic series and the second factors form a geometric series. What technique could I use to derive a formula for the sum of this series? Could I let S denote the sum of the general geometric series of n terms. Multiply S by r and write the series corresponding to the difference between S and rS. How would I use this expression to deducde the formula for the sum of the general geometric series?
I think you'll find this useful:
PlanetMath: arithmetic-geometric series
Hello, Keith!
The general arithmetic-geometric series of terms is a series of the form:
. .
where the first factors of each term form an arithmetic series
and the second factors form a geometric series.
What technique could I use to derive a formula for the sum of this series?
Could I let denote the sum of the general geometric series of terms?
Multiply by and then write the series corresponding
. . to the difference between and ? . Yes ... a great game plan!
How would I use this to deduce the formula for the sum of the general series?
The original series:
. . .
Multiply by
. . .
Subtract: .
. . The geometric series has first term , common ratio , and terms.
. . . . Its sum is: .
So we have: .
. .
Therefore: .