Thread: The general arithmetic-geometric series

I am looking for the general arithmetic-geometric series of n terms in the form of: 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 from a geometric series. I am looking for a website that shows this step by step so I can be able to explain this in my education class, I am in calculus and we have not covered this material as of yet. I looked in my calculus text and the method to derive this formula is not listed. Can someone point me in the right direction.

Hello, Keith!

This is not a pleasant problem . . . especially without LaTeX!

I am looking for the general arithmetic-geometric series of n terms in the form of:

.S . = . a = (a+d)r + (a+2d)r² + (a+3d)r³ + ... + [a + (n-1)d]r^{n-1}

We start with:
. . S . = . a + (a+d)r + (a+2d)r² + (a+3d)r³ + . . . + [a + (n-1)d]r^{n-1}

Multiply by r:
. .rS . = . . . . . ar . + . .(a+d)r² + (a+2d)r³ + . . . + [a + (n-2)d]^{n-1} + [a + (n-1)d]r^n

Subtract:
. . S - rS . = . a + dr + dr² + dr³ + . . . + dr^{n-1} - [a + (n-1)d]r^n
. . . . . . . . . . . . . \_________________________/


The middle group is a geometric series with first term dr,
. . common ratio r, and n terms.

. . . . . . . . . . . 1 - r^n
Its sum is: . dr ---------
. . . . . . . . . . . .1 - r

. . . . . . . . . . . . . . . . . . . . . . . .1 - r^n
So we have: . (1 - r)S . = . a + dr -------- - [a + (n-1)d]r^n
. . . . . . . . . . . . . . . . . . . . . . . . .1 - r


I'll let you solve for S . . . I need a nap!

Soroban is this correct?
(1-r)S = a(1-r^n)/1-r + dr(1-r^n-1)/(1-r)^2 - [a = (n-1)d]r^n/(1-r)

Thank you,
Keith Stevens