💰
Escape Room · Annuities
The Nest Egg
Coastal Savings Bank · Retirement Portal · Statement deadline tonight
← Notes
Verifications complete: 0 of 6 0%
1
2
3
4
5
6
📍 Scene Setup Josh's nan has been saving for retirement through the Coastal Savings Bank. Tonight the bank's retirement portal generates her annual statement, but the system requires all calculations to be manually verified before it will finalise the document.

Josh has offered to run the numbers. Six verifications, and Nan gets her statement.

📋 Nan's Retirement Account

Opening balance (V₁)
$1000
Annual interest rate
10%
Annual deposit (d)
$500
Recurrence: Vₙ₊₁ = 1.1 × Vₙ + 500  ·  Interest earned = Vₙ × 0.10 each year
1
Year 1, Balance After First Deposit
Nan's account opens with V₁ = $1000. After one year of 10% growth plus a $500 deposit, what is V₂?
Use: V₂ = 1.1 × V₁ + 500
💡 V₂ = 1.1 × 1000 + 500 = 1100 + 500 = ?
✅ V₂ = $1600. The $1000 grew by $100 in interest, then $500 was deposited.
2
Year 2, Continuing the Growth
Using V₂ = $1600, apply the same recurrence to find V₃.
V₃ = 1.1 × V₂ + 500. Notice the interest amount is larger this year.
💡 V₃ = 1.1 × 1600 + 500 = 1760 + 500 = ?
✅ V₃ = $2260. Interest earned this year: $160, up from $100 last year because the balance is larger.
3
Year 3, One More Year
Find V₄ using V₃ = $2260.
Same recurrence: V₄ = 1.1 × V₃ + 500
💡 V₄ = 1.1 × 2260 + 500 = 2486 + 500 = ?
✅ V₄ = $2986. Three years of growth: $1000 → $1600 → $2260 → $2986. The jumps are getting bigger.
4
Second Account, Different Rates
Nan's friend opens an account with V₁ = $500, earns 20% per year, and deposits $100 annually.

What is V₂ for this account?
New recurrence: V₂ = 1.2 × 500 + 100
💡 R = 1.2 (for 20% interest). V₂ = 1.2 × 500 + 100 = 600 + 100 = ?
✅ V₂ = $700. Higher interest rate means faster growth, even with smaller deposits.
5
Total Interest Earned
Back to Nan's account. Over 3 years she deposited:
$1000 (opening) + 3 × $500 (annual deposits) = $2500 total deposited.

After 3 years the balance is $2986. How much total interest did the account earn?
Interest = final balance − total deposited
💡 $2986 − $2500 = ?   (total deposited = $1000 + $500 + $500 + $500)
✅ $486 in interest earned. Nan put in $2500 and got back $2986, the extra $486 came purely from compound interest.
6
Loan vs Savings, Spot the Difference
Consider the same starting values, $1000, R = 1.1, amount = $500, but now $500 is being repaid (like a loan) rather than deposited.

What is V₂ for this loan?
Loan recurrence: V₂ = 1.1 × 1000 − 500. Notice the minus, it's the only difference!
💡 Loan: V₂ = 1.1 × 1000 − 500 = 1100 − 500 = ?   Compare to savings V₂ = $1600, same numbers, completely different result!
✅ Loan V₂ = $600 vs savings V₂ = $1600. Same starting point, same R, same $500, one tiny sign change makes all the difference.
💰
Statement Generated!
All 6 verifications passed. The Coastal Savings Bank portal has finalised Nan's retirement statement. Her nest egg has grown from $1000 to $2986 in three years, $486 of which is pure interest.

The difference between + and − might be one character, but it's the difference between building wealth and paying down debt. 🌟
Vₙ₊₁ = R·Vₙ + d  (savings)
Vₙ₊₁ = R·Vₙ − d  (loan)
Interest = final − deposited
← Back to Notes