Quick questions on Annuities and future value: deriving the formula and applying it to regular savings

An annuity is a sequence of equal payments made at regular intervals. In the ordinary annuity model used in Maths Advanced, each payment is made at the end of a compounding period, and interest is credited at the same rate per period that compounding occurs.

Track when each payment is made and how many full periods it earns interest before time $n$.

Solve for $M$ when the target balance $A$ is given:

Some questions credit interest at the start of each period instead of after, or count the balance just before the next deposit. The number of compounding periods for each deposit changes by $1$. Read the question carefully and either reuse the geometric-series derivation or multiply $A$ by an extra factor of $(1 + r)$.

Deposit $\$500$ at the end of each quarter into an account paying $8\%$ per annum compounded quarterly for $5$ years.

You want $\$100000$ in $20$ years. Account pays $6\%$ per annum compounded monthly. Monthly deposit?

Deposit $\$1000$ at the end of each year at $5\%$ per annum compounded annually. How many years to reach $\$25000$?

For a small case, list the contributions and check the formula. Three deposits of $\$100$ at the end of each year at $10\%$ per annum:

In an ordinary annuity the last payment earns zero interest. Off-by-one errors are common; always check with a small case.

Convert to the per-period rate first.

Total deposited is $M n$. Future value $A$ is larger by the interest earned. Some questions ask for the interest, which is $A - M n$.

When solving for $M$, the formula is $M = \frac{A r}{(1 + r)^n - 1}$, not $M = \frac{A}{(1 + r)^n}$ (that is a single lump-sum discount).

Annuity due payments are made at the start of each period and earn one extra period of interest, multiplying the future value by $(1 + r)$. The default in Maths Advanced is the ordinary annuity unless stated otherwise.