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What even is this? An annuity is a savings or investment account where you make regular deposits, and interest compounds on the growing balance at the same time. It's the mirror image of a reducing balance loan: instead of paying down debt, you're building up wealth. The only difference in the formula is a + instead of a −.
Section 1 · Loans vs Annuities: Spot the Difference
Reducing Balance Loan
Vₙ₊₁ = R·Vₙ − d
You owe money. Each period, interest is added to what you owe, then you subtract a repayment. The balance decreases over time (if repayments are large enough).
Annuity (Savings)
Vₙ₊₁ = R·Vₙ + d
You own money. Each period, interest is earned on your balance, then you add a deposit. The balance increases every period, growing faster over time.
The maths is identical, same recurrence structure, same R = 1 + r, the only difference is the sign on d. Minus for loans, plus for savings. If you mix them up, your balance will head the wrong direction.
Section 2 · The Recurrence Relation
Vn+1 = R × Vn + d
Vₙ = account balance at period n
R = 1 + r = growth multiplier (e.g. 10% → R = 1.1)
d = regular deposit each period (positive)
V₁ = starting balance (could be 0)
Section 3 · Generating the Balance Step by Step
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Example: V₁ = 1000, R = 1.1, d = 500 (find V₂, V₃, V₄)
1
V₁ = $1000 (starting balance, given)
2
V₂ = 1.1 × 1000 + 500 = 1100 + 500 = $1600
3
V₃ = 1.1 × 1600 + 500 = 1760 + 500 = $2260
4
V₄ = 1.1 × 2260 + 500 = 2486 + 500 = $2986
Notice the jumps get bigger each period: +$600, +$660, +$726. That's the compounding effect, interest is being earned on a larger balance each time, so the growth accelerates. This is why starting early matters so much in real life.
Section 4 · Total Interest Earned
💡 Key formula, interest earned
Total interest = Final balance − Total deposited
Add up all the deposits (including V₁ if it was a deposit). Subtract that from the final balance. What's left is the interest the account earned for you.
Example (from above): V₁ = $1000 (initial), then 3 deposits of $500. Total deposited = $1000 + 3 × $500 = $2500. Final balance = $2986. Interest earned = $2986 − $2500 = $486.
Example (from above): V₁ = $1000 (initial), then 3 deposits of $500. Total deposited = $1000 + 3 × $500 = $2500. Final balance = $2986. Interest earned = $2986 − $2500 = $486.
Section 5 · Worked Example
💰 Nan's Nest Egg
Josh's nan opens a savings account with $1000 and adds $500 at the end of each year. The account earns 10% interest per year. She wants to know the balance after 3 years and how much of that is interest.
Step 1 · Write the recurrence
Identify R and d, then write the relation.
Starting balance: V₁ = $1000
Interest rate: r = 10% → R = 1.1
Regular deposit: d = $500
Vₙ₊₁ = 1.1 × Vₙ + 500, V₁ = 1000
Interest rate: r = 10% → R = 1.1
Regular deposit: d = $500
Vₙ₊₁ = 1.1 × Vₙ + 500, V₁ = 1000
Step 2 · Generate balances
What is the balance at the end of each year?
| Year | Opening | Interest (×10%) | Deposit | Closing (Vₙ₊₁) |
|---|---|---|---|---|
| 1→2 | $1000 | +$100 | +$500 | $1600 |
| 2→3 | $1600 | +$160 | +$500 | $2260 |
| 3→4 | $2260 | +$226 | +$500 | $2986 |
| After 3 years of deposits: | $2986 | |||
Step 3 · Calculate total interest earned
How much did the account earn in interest over the 3 years?
Total deposited = $1000 (initial) + 3 × $500 = $1000 + $1500 = $2500
Final balance = $2986
Interest earned = $2986 − $2500 = $486
The interest itself grew each year ($100 → $160 → $226) because the balance it was calculated on kept growing.
Final balance = $2986
Interest earned = $2986 − $2500 = $486
The interest itself grew each year ($100 → $160 → $226) because the balance it was calculated on kept growing.
Section 5.5 · Try it halfway
Step 1 is done for you. Fill in the gaps in Steps 2 and 3, then reveal to check.
Section 6 · Practice Questions
Tap to reveal the worked answer.
Question 1
An account starts at V₁ = $200, earns 20% per period, and receives a deposit of $100 each period.
Write the recurrence and find V₂.
Write the recurrence and find V₂.
▼
R = 1.2, d = 100
Recurrence: Vₙ₊₁ = 1.2Vₙ + 100, V₁ = 200
V₂ = 1.2 × 200 + 100 = 240 + 100 = $340
Recurrence: Vₙ₊₁ = 1.2Vₙ + 100, V₁ = 200
V₂ = 1.2 × 200 + 100 = 240 + 100 = $340
✓ Vₙ₊₁ = 1.2Vₙ + 100, V₂ = $340
Question 2
Continue from Q1 (V₂ = $340, same recurrence). Find V₃.
▼
V₃ = 1.2 × 340 + 100 = 408 + 100 = $508
✓ V₃ = $508
Question 3
A savings plan starts with $2000, grows at 15% per period, and receives a $300 deposit each period.
Write the recurrence and find the balance after the first period (V₂).
Write the recurrence and find the balance after the first period (V₂).
▼
R = 1.15, d = 300
Recurrence: Vₙ₊₁ = 1.15Vₙ + 300, V₁ = 2000
V₂ = 1.15 × 2000 + 300 = 2300 + 300 = $2600
Recurrence: Vₙ₊₁ = 1.15Vₙ + 300, V₁ = 2000
V₂ = 1.15 × 2000 + 300 = 2300 + 300 = $2600
✓ Vₙ₊₁ = 1.15Vₙ + 300, V₂ = $2600
Question 4, Total interest
Using Q1, Q2 (V₁ = $200, V₃ = $508, two deposits of $100 made):
How much total interest did the account earn?
How much total interest did the account earn?
▼
Total deposited = $200 (initial) + $100 + $100 = $400
Final balance = $508
Interest earned = $508 − $400 = $108
Final balance = $508
Interest earned = $508 − $400 = $108
✓ $108 earned in interest
Question 5, Pension (withdrawal)
A pension account starts at $1000 and earns 20% per period. Each period, $200 is withdrawn.
What is V₂? What do you notice?
What is V₂? What do you notice?
▼
This is a withdrawal, so d is subtracted: Vₙ₊₁ = 1.2Vₙ − 200
(This is actually a reducing balance recurrence, same structure as a loan!)
V₂ = 1.2 × 1000 − 200 = 1200 − 200 = $1000
The balance stays exactly at $1000! The 20% interest earned ($200) is exactly cancelled by the $200 withdrawal. This is a sustainable pension, it never runs out at this withdrawal rate.
(This is actually a reducing balance recurrence, same structure as a loan!)
V₂ = 1.2 × 1000 − 200 = 1200 − 200 = $1000
The balance stays exactly at $1000! The 20% interest earned ($200) is exactly cancelled by the $200 withdrawal. This is a sustainable pension, it never runs out at this withdrawal rate.
✓ V₂ = $1000. Withdrawals = interest earned → balance stays flat permanently.
The Nest Egg, Escape Room
Nan's retirement savings need to be verified before the bank portal closes. 6 challenges on annuities, building balances, comparing loans vs savings, and calculating total interest earned.
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