FINANCIAL APPLICATIONS OF GEOMETRIC SEQUENCES AND SERIES
This question bank contains 12 questions covering financial applications of geometric sequences and series, distributed across different paper types according to IB AAHL curriculum standards.
📌 Multiple Choice Questions (5 Questions)
MCQ 1. $3000 is invested at 4% per annum, compounded annually, for 5 years. What is the final amount?
A) $3649.96 B) $3600.00 C) $3750.00 D) $3500.00
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Solution:
Using compound interest formula: \(FV = PV(1 + \frac{r}{100})^n\)
\(FV = 3000(1 + \frac{4}{100})^5 = 3000(1.04)^5\)
\(FV = 3000 \times 1.2167 = \$3649.96\)
✅ Answer: A) $3649.96
MCQ 2. A car worth $18,000 depreciates at 10% per annum. What is its value after 3 years?
A) $13,122 B) $14,400 C) $12,600 D) $15,120
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Solution:
Using depreciation formula: \(FV = PV(1 – \frac{d}{100})^n\)
\(FV = 18000(1 – \frac{10}{100})^3 = 18000(0.9)^3\)
\(FV = 18000 \times 0.729 = \$13,122\)
✅ Answer: A) $13,122
MCQ 3. What is the effective annual rate for 6% compounded quarterly?
A) 6.00% B) 6.14% C) 6.09% D) 6.25%
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Solution:
Using EAR formula: \(EAR = (1 + \frac{r}{100k})^k – 1\)
For quarterly compounding: k = 4, r = 6%
\(EAR = (1 + \frac{6}{400})^4 – 1 = (1.015)^4 – 1\)
\(EAR = 1.0614 – 1 = 0.0614 = 6.14\%\)
✅ Answer: B) 6.14%
MCQ 4. An investment grows from $2000 to $2662 in 4 years with annual compounding. What is the interest rate?
A) 7.5% B) 8.0% C) 7.3% D) 8.3%
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Solution:
Using: \(FV = PV(1 + \frac{r}{100})^n\)
\(2662 = 2000(1 + \frac{r}{100})^4\)
\((1 + \frac{r}{100})^4 = 1.331\)
\(1 + \frac{r}{100} = (1.331)^{0.25} = 1.075\)
Therefore: \(r = 7.5\%\)
✅ Answer: A) 7.5%
MCQ 5. Which gives the highest return on $1000 invested for 2 years?
A) 5% simple interest B) 4.8% compounded annually
C) 4.7% compounded quarterly D) 4.6% compounded monthly
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Solution:
Calculate each option:
A) Simple: \(1000(1 + \frac{5 \times 2}{100}) = \$1100\)
B) Annual: \(1000(1.048)^2 = \$1098.30\)
C) Quarterly: \(1000(1.01175)^8 = \$1098.08\)
D) Monthly: \(1000(1.003833)^{24} = \$1097.95\)
✅ Answer: A) 5% simple interest
📌 Paper 2 Questions (Calculator Allowed) – 6 Questions
Paper 2 – Q1. Sarah invests $7500 in an account that pays 3.2% per annum, compounded semi-annually.
(a) Calculate the value of her investment after 8 years. [4 marks]
(b) How long will it take for her investment to double? [3 marks]
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Solution:
(a) Value after 8 years:
Given: PV = $7500, r = 3.2% p.a., k = 2, n = 8 years
\(FV = 7500(1 + \frac{3.2}{200})^{16}\)
\(FV = 7500(1.016)^{16} = 7500 \times 1.2952 = \$9714\)
(b) Time to double:
Need: \(15000 = 7500(1.016)^{2n}\)
\((1.016)^{2n} = 2\)
\(2n \ln(1.016) = \ln(2)\)
\(n = \frac{\ln(2)}{2\ln(1.016)} = \frac{0.693}{0.0318} = 21.8\) years
✅ Answer: (a) $9714; (b) 21.8 years
Paper 2 – Q2. A laptop costs $1200 and depreciates at 20% per annum.
(a) Find its value after 3 years. [3 marks]
(b) After how many complete years will its value fall below $300? [4 marks]
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Solution:
(a) Value after 3 years:
\(FV = 1200(1 – 0.2)^3 = 1200(0.8)^3\)
\(FV = 1200 \times 0.512 = \$614.40\)
(b) When value < $300:
\(1200(0.8)^n < 300\)
\((0.8)^n < 0.25\)
\(n \ln(0.8) < \ln(0.25)\)
\(n > \frac{\ln(0.25)}{\ln(0.8)} = \frac{-1.386}{-0.223} = 6.22\)
Therefore after 7 complete years
✅ Answer: (a) $614.40; (b) 7 years
Paper 2 – Q3. Emma takes a car loan of $25,000 at 4.8% per annum, compounded monthly, for 5 years.
(a) Calculate her monthly payment (EMI). [4 marks]
(b) Find the total amount she pays over 5 years. [2 marks]
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Solution:
(a) Monthly EMI:
Given: P = $25,000, r = 4.8% p.a., n = 60 months
Monthly rate = 4.8/1200 = 0.004
\(EMI = \frac{25000 \times 0.004 \times (1.004)^{60}}{(1.004)^{60}-1}\)
\(EMI = \frac{100 \times 1.2712}{0.2712} = \$468.79\)
(b) Total amount paid:
Total = $468.79 × 60 = $28,127.40
✅ Answer: (a) $468.79; (b) $28,127.40
Paper 2 – Q4. Compare two investment options for $10,000 over 6 years:
Option A: 5.5% compounded annually
Option B: 5.2% compounded monthly
(a) Calculate the final value for each option. [4 marks]
(b) Which option is better and by how much? [2 marks]
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Solution:
(a) Calculate both options:
Option A (Annual):
\(FV_A = 10000(1.055)^6 = 10000 \times 1.3788 = \$13,788\)
Option B (Monthly):
\(FV_B = 10000(1 + \frac{5.2}{1200})^{72}\)
\(FV_B = 10000(1.004333)^{72} = 10000 \times 1.3737 = \$13,737\)
(b) Better option:
Option A is better by $13,788 – $13,737 = $51
✅ Answer: (a) A: $13,788; B: $13,737; (b) Option A by $51
Paper 2 – Q5. A company’s revenue was $2.5 million in 2020. Due to inflation, the real value of this revenue decreases by 3% each year.
(a) What is the real value of this revenue in 2025? [3 marks]
(b) In which year will the real value first fall below $2 million? [4 marks]
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Solution:
(a) Real value in 2025:
Time period: 2025 – 2020 = 5 years
Real value = \(2.5(1 – 0.03)^5 = 2.5(0.97)^5\)
Real value = \(2.5 \times 0.8587 = \$2.147\) million
(b) When real value < $2 million:
\(2.5(0.97)^n < 2\)
\((0.97)^n < 0.8\)
\(n \ln(0.97) < \ln(0.8)\)
\(n > \frac{\ln(0.8)}{\ln(0.97)} = \frac{-0.223}{-0.0305} = 7.31\)
Therefore in year: 2020 + 8 = 2028
✅ Answer: (a) $2.147 million; (b) 2028
Paper 2 – Q6. Marcus invests $8000 at an unknown interest rate, compounded quarterly. After 4 years, his investment is worth $9856.
(a) Find the annual interest rate. [4 marks]
(b) What would be the effective annual rate? [2 marks]
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Solution:
(a) Finding interest rate:
Given: PV = $8000, FV = $9856, n = 4 years, k = 4
\(9856 = 8000(1 + \frac{r}{400})^{16}\)
\((1 + \frac{r}{400})^{16} = 1.232\)
\(1 + \frac{r}{400} = (1.232)^{1/16} = 1.0133\)
\(\frac{r}{400} = 0.0133\), so \(r = 5.32\%\)
(b) Effective annual rate:
\(EAR = (1 + \frac{5.32}{400})^4 – 1\)
\(EAR = (1.0133)^4 – 1 = 1.0544 – 1 = 5.44\%\)
✅ Answer: (a) 5.32%; (b) 5.44%
📌 Paper 3 Question (Extended Response) – 1 Question
Paper 3 – Extended Question. Lisa is planning for retirement and considering two financial strategies over 30 years:
Strategy A: Invest $500 monthly in an account paying 6% per annum, compounded monthly.
Strategy B: Invest $1500 quarterly in an account paying 6.2% per annum, compounded quarterly.
(a) Calculate the future value of Strategy A after 30 years. [4 marks]
(b) Calculate the future value of Strategy B after 30 years. [4 marks]
(c) Which strategy gives better returns and by how much? [2 marks]
(d) Calculate the total amount invested in each strategy. [2 marks]
(e) If inflation averages 2.5% per year, what is the real value of the better strategy in today’s purchasing power? [3 marks]
(f) Recommend which strategy Lisa should choose, justifying your answer with at least three financial considerations. [5 marks]
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Complete Solution:
(a) Strategy A Future Value:
This is a monthly annuity: PMT = $500, r = 6% p.a., n = 360 months
Monthly rate = 6/1200 = 0.005
Using annuity formula: \(FV = PMT \times \frac{(1+i)^n – 1}{i}\)
\(FV_A = 500 \times \frac{(1.005)^{360} – 1}{0.005}\)
\(FV_A = 500 \times \frac{6.023 – 1}{0.005} = 500 \times 1004.6 = \$502,300\)
(b) Strategy B Future Value:
Quarterly annuity: PMT = $1500, r = 6.2% p.a., n = 120 quarters
Quarterly rate = 6.2/400 = 0.0155
\(FV_B = 1500 \times \frac{(1.0155)^{120} – 1}{0.0155}\)
\(FV_B = 1500 \times \frac{6.317 – 1}{0.0155} = 1500 \times 343.0 = \$514,500\)
(c) Better strategy:
Strategy B is better by $514,500 – $502,300 = $12,200
(d) Total invested:
Strategy A: $500 × 360 = $180,000
Strategy B: $1500 × 120 = $180,000
(e) Real value with inflation:
Better strategy (B) real value = \(\frac{514,500}{(1.025)^{30}}\)
Real value = \(\frac{514,500}{2.098} = \$245,354\)
(f) Recommendation:
Financial Considerations:
1. Higher Returns: Strategy B provides $12,200 more despite equal total investment
2. Higher Interest Rate: 6.2% vs 6% compounds significantly over 30 years
3. Quarterly Discipline: Larger quarterly payments may be easier to manage than monthly commitments
4. Flexibility: Quarterly payments allow for better cash flow management
5. Compounding Benefit: Higher rate overcomes the less frequent compounding
Caution: Strategy B requires disciplined saving of larger amounts quarterly, which may be challenging if income is irregular.
✅ Final Answers:
(a) Strategy A: $502,300
(b) Strategy B: $514,500
(c) Strategy B better by $12,200
(d) Both invest $180,000 total
(e) Real value: $245,354
(f) Recommend Strategy B for higher returns