question about geometric series?

500 * Σ (t from 1 to 180) 1 / (1+x)^t = 99000.

Σ (t from 1 to 180) 1 / (1+x)^t = 99000/500 = 198.

For the moment, let r = 1 / (1+x). It won't really matter.

Σ (t from 1 to 180) r^t = 198

r * Σ (t from 0 to 179) r^t = 198

r * (r^180 - 1) / (r - 1) = 198

r * (r^180 - 1) = 198(r - 1)

r^181 - r = 198r - 198

r^181 - 199r + 198 = 0

This has one solution we know of... r = 1. But that is a false solution (note the division by r-1=0). Rational roots theorem (±divisors of 198 are only possibilities) produced nothing else.

Maple "barfed" when I tried to get it to find all the roots.

You might get an approximate (I don't know how good) answer by interpreting the sum as an integral:

Σ (t from 1 to 180) r^t = 198

∫ (from 0 to 180) r^t dt = 198 (r is fixed, write r^t as e^(t ln r); this is right endpoint approximation)

∫ (from 1 to 181) r^t dt = 198 (this is left endpoint approximation)

and solve for r, then x from there - but that looks troublesome too.

Hope this helps.

♣ ♣