Repeated-deposit sequence models and timing
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| deposit/dɪˈpɒzɪt/ | tiền gửi |
Which timeline does the sum describe?
- Two equal payments can earn different amounts of interest if one is paid earlier.
- This lesson studies deposit 存款: An amount added to an account at a stated time.
The method
- 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.
Why is the final end-of-period deposit not given an interest period in B_n?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
With end-of-period deposits, B_n=d(1+r+...+r^(n-1)): the exponents count the periods EACH deposit has been sitting. Giving the last deposit interest would double-count time.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The second balance is 1.05×100+100=205.
Where it goes wrong
- 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.
Not true: Beginning- and end-of-period deposits give the same final balance. A deposit at the start of a period earns one more period of interest, so the final balance is larger.
Find B₃ in that model. Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The third balance is 1.05×205+100=315.25.
A deposit at the start of a period earns one more period of interest, so the final balance is larger.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. A deposit at the start of a period earns one more period of interest, so the final balance is larger.
Applying it
- 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.
Find the first whole period with B_n≥500.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
B₄=431.0125 is below 500 and B₅=552.563125 is above it; increasing balances give the first crossing at 5.
Match each deposit-model fact to its statement.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Deposits at period START shift every balance by one growth factor: C_n=rB_n, so timing must be stated before computing.
In your course
- AQA A-level Mathematics (7357), specification section D.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Deposit: An amount added to an account at a stated time.
- Define the balance immediately after a stated event and give the initial balance.
- Watch for: do not give the final end-of-period deposit an extra interest period.