Quick questions on Simple and compound interest, future value and present value for HSC Maths Advanced

Simple interest is paid only on the original principal. After $n$ time periods at per-period rate $r$,

Compound interest is added to the principal at the end of each period and itself earns interest in every subsequent period. After $n$ compounding periods at per-period rate $r$,

The present value $P$ is the amount you must invest today to grow to a target $A$ in $n$ periods. Rearranging the compound interest formula,

To make the discount tangible, take the present-value calculation from the worked example below: how much must you invest today to have $20000 in $5$ years at $6\%$ per annum compounded annually? Rather than divide by $(1.06)^5$ in one move, walk the future amount back one year at a time. Each year you step toward today, you divide by $1.06$ once.

The effective annual rate makes different compounding frequencies comparable. If the nominal annual rate is $R$ compounded $m$ times per year,