Hello, Keith!

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

I am looking for the general arithmetic-geometric series ofnterms 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 byr:

. .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 termdr,

. . common ratior, andnterms.

. . . . . . . . . . . 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 letyousolve forS. . . I need a nap!