Quick questions on Annuity Investments and Superannuation - TCE Mathematics Applications (Tasmania)

For many periods, iterating by hand is slow, so an explicit future-value formula is used. For an ordinary annuity (contributions at the end of each period), $$FV = d\,\frac{(1 + i)^n - 1}{i},$$where$d$is the regular contribution,$i$is the per-period rate, and$n$is the number of contributions. If the contributions are made at the start of each period (an annuity in advance), multiply the whole expression by an extra$(1 + i)$, because every deposit then earns one more period of interest. Any starting balance$V_0$grows separately as$V_0(1 + i)^n$ and is added on.

A common task gives a savings target and asks for the required regular deposit. Rearrange the future-value formula to make $d$ the subject by dividing the target $FV$ by the annuity factor $\frac{(1 + i)^n - 1}{i}$ (times $(1 + i)$ for an annuity in advance). This is exactly the algebra in the second exam answer above, and the marker rewards showing the rearrangement, not just the final number.