What will the future value be if you open the account with $1,000 today, and then make the $100 deposits at the end of each year?

Question

BlackTom AI · Accepted Answer

Future value = future value of the $1,000 lump sum compounded for 10 years plus the future value of a 10-year ordinary annuity of $100 at 8% per year.

Relevant concepts:
- Future value of a lump sum: FV = PV × (1 + r)^n.
- Future value of an ordinary annuity (payments at end of each period): FVann = PMT × [((1 + r)^n − 1) / r].
Here r = 8% = 0.08, n = 10, PV = $1,000, PMT = $100.

Step-by-step calculation:
1. Compute (1 + r)^n = 1.08^10.
   1.08^10 = 2.158924997 (carry sufficient digits for accuracy).
2. Future value of the $1,000 lump sum:
   FV_lump = 1000 × 1.08^10 = 1000 × 2.158924997 = 2,158.924997 ≈ 2,158.93.
3. Future value factor for the annuity:
   ((1 + r)^n − 1) / r = (2.158924997 − 1) / 0.08 = 1.158924997 / 0.08 = 14.48656246.
4. Future value of the $100 annuity:
   FV_ann = 100 × 14.48656246 = 1,448.656246 ≈ 1,448.66.
5. Total future value:
   FV_total = FV_lump + FV_ann = 2,158.924997 + 1,448.656246 = 3,607.581243 ≈ 3,607.58.

Why this answer is correct:
The calculation correctly compounds the initial deposit for all 10 years and treats the $100 deposits as end-of-year payments (ordinary annuity), compounding each deposit for the remaining years. The numerical rounding yields $3,607.58, matching the authoritative answer.

Common incorrect approaches (and why they’re wrong):
- Treating the $100 deposits as an annuity due (beginning of year) would give a higher FV.
- Omitting compounding of the initial $1,000 or failing to use the annuity FV formula would understate the result.
- Using the wrong rate or number of periods produces different (incorrect) totals.

Final confirmation: The computed future value is $3,607.58, which matches the provided answer.

What will the future value be if you open the account with $1,000 today, and then make the $100 deposits at the end of each year?数值题

类似问题

Study the screenshot above.Miguel invests $1500 every month for 5 years with an annual interest rate of 8.23%. What formula is needed to calculate what Miguel's investment is worth at the end of 5 years? Show as a positive number.

Question13 Suppose you receive $1,100 per year for 5 years, starting with your first payment at the end of year 1, earning a return of 6.5%. What is the FV of these cash flows approximately equal to?Select one alternative: $4,571 $10,392 $6,263 $14,441 ResetMaximum marks: 3 Flag question undefined

You plan to save for your child's education by depositing $880 at the end of each month into an account that earns an annual interest rate of 8%, compounded monthly. If you plan to invest for 12 years, calculate the value at the end of this period. Please write the answer in two decimal places.

Your goal is to accumulate $149,904 in nine years' time. What monthly deposit must you make into a savings account to reach your goal? Interest rates are 22% per annum, compounded monthly. Please write the answer in two decimal places.

Andrew would like to accumulate $15,000 in his savings account at the end of 4 years. If effective annual interest rate is 11%. What would be the amount of equal deposit at the end of each year for 4 years.

You want to have $1 million to use for retirement in 35 years. If you earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单