Reinvesting the interest (generalized version)?

Question

If I deposit \\$1 at $t=0$ into an account which credits interest at the end of each year at a force of interest $\\delta_t$ (assume it's integrable.) Then, if I reinvest the interest at an annual effective rate interest $j,$ what'd be the total accumulated amount?To generalize further what if the first interest is given by $f(t)$ and reinvested interest is $g(t)?$

Answer 1

You invest 1 dollar today.

in 1 year you have 1+i

in Two years you have (1+i)^2 = 1 +2i +i^2 the one dollar has earned interest twice and the interest has earned interest once.

in 3 years you will have (1+i)^3

at year t, you will have (1+i)^t Total interest received will equal (1+i)^t -1.

Total interest on the principal is t*i One dollar earned i for t years.

Total interest on interest is therefore (1+i)^t - 1 - ti.

We can do some other neat manipulations with this if you like