A financial plan grows money in two ways at once. A lump sum you put in and leave alone, and regular contributions you keep adding every period. Both earn compound interest. Planning is just choosing the tool that matches the question.
d = the payment each period · i = rate per period (decimal) · n = number of periods. To reach a goal, rearrange for d: d = AFV × i ÷ [(1 + i)ⁿ − 1].
$200 deposited at the end of each year for 3 years, earning 10% p.a. Find the total.
| Deposit (end of year) | Grows for | Value at end |
|---|---|---|
| Year 1 | 2 years | 200 × 1.10² = $242 |
| Year 2 | 1 year | 200 × 1.10 = $220 |
| Year 3 | 0 years | 200 × 1 = $200 |
| Total | $662 |
The formula just adds these three up in one line for you.
• Use A = P(1 + i)ⁿ for a one-off lump sum, and AFV for repeated payments. Don't mix them.
• i is the rate per period, match it to how often you pay.
• "Reach a goal" questions give you A and ask for d, rearrange, don't guess.
• Round only at the very end.
Same $200 deposit each year, but the earliest one earns the most interest (the teal cap), because it grows for longest. Add the three: $662.
Read the question: is it one lump sum, or a repeating payment? That picks your formula.
Don't read yet, just have a go in your head:
d = 200, i = 0.10, n = 3 → AFV = 200 × [(1.10)³ − 1] ÷ 0.10 = 200 × [1.331 − 1] ÷ 0.10 = 200 × 3.31 = $662
d = 200, i = 0.10, n = 3 → AFV = 200 × [(1.10)³ − 1] ÷ 0.10 = 200 × [? − 1] ÷ 0.10 = 200 × ?
$100 a year for 2 years at 10% p.a. Write d, i, n, then find AFV. Check below.
You want $2,200 in 2 years, depositing the same amount each year at 20% p.a. How much must you deposit each year?
In one sentence, out loud: how do you know whether to use the lump-sum formula or the payments formula? If you can say it, you've got it.