Repeated-deposit sequence models and timing
| English | Português |
|---|---|
| deposit/dɪˈpɒzɪt/ | depósito |
Two equal payments can earn different amounts of interest if one is paid earlier. Which timeline does the sum describe?
- Two equal payments can earn different amounts of interest if one is paid earlier. Which timeline does the sum describe?
- This lesson studies deposit 存款: An amount added to an account at a stated time.
Choose the mathematical structure
- Define the balance immediately after a stated event and give the initial balance. With growth factor r and an end-of-period deposit d, B₀=0 and B_n=rB_(n−1)+d. The nth balance is d(1+r+…+r^(n−1))=d(r^n−1)/(r−1) for r≠1. For r=1, use B_n=nd. The period count n is a nonnegative integer.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines deposit?
An amount added to an account at a stated time.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Use an illustrative growth rate of 5% per period and deposit 100 units at each period end. B₁=100, B₂=1.05×100+100=205 and B₃=1.05×205+100=315.25. The first deposit earns interest for n−1 periods, while the final one earns none at the time B_n is measured. Thus B_n=100(1.05^n−1)/0.05. For a target of at least 500, B₄=431.0125<500 and B₅=552.563125≥500, so 5 periods first suffice. The balance exceeds the five deposits totalling 500 by 52.563125 units. If the same deposits are made at each period start and the balance is measured at the period end, C_n=1.05(B_n); hence C₃=331.0125, not 315.25.
Repeated-deposit sequence models and timing
Define the balance immediately after a stated event and give the initial balance
Trace each payment through the stated compounding timeline.
Find B₂ in the illustrative end-of-period model.
The second balance is 1.05×100+100=205.
Test a tempting shortcut
- Do not give the final end-of-period deposit an extra interest period. A balance includes both deposits and growth; it is not the interest alone. An exact target and an at-least target need different integer checks. A quoted annual rate cannot simply be used as a per-month factor without an appropriate conversion.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Beginning- and end-of-period deposits always give the same final balance. This claim is false. Explain which definition or assumption it violates.
Find B₃ in that model.
The third balance is 1.05×205+100=315.25.
Beginning- and end-of-period deposits always give the same final balance.
Do not give the final end-of-period deposit an extra interest period. A balance includes both deposits and growth; it is not the interest alone. An exact target and an at-least target need different integer checks. A quoted annual rate cannot simply be used as a per-month factor without an appropriate conversion.
Interpret a new situation
- The illustration assumes a fixed per-period rate, equal deposits, no fees or withdrawals and the stated compounding/payment times. If these change, refine the recurrence: B_n=(1+i_n)B_(n−1)+d_n−w_n for end-of-period deposits/withdrawals. Keep a separate ledger of payments to identify growth. Compare adjacent period balances before claiming the first target crossing; here all balances are nonnegative and B_n−B_(n−1)=0.05B_(n−1)+100>0.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the first whole period with B_n≥500.
B₄=431.0125 is below 500 and B₅=552.563125 is above it; increasing balances give the first crossing at 5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · D. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An amount added to an account at a stated time. Choose the relationship, show the method, check its assumptions and interpret the result.