IB Mathematics SL 1.4 Financial applications of geometric sequences and series AI SL Paper 1- Exam Style Questions- New Syllabus
Question
Note: Provide all financial answers correct to two decimal places.
Myte initiates a long-term savings strategy by depositing \( 1500 \) euros (\( \text{EUR} \)) into an investment account at the end of every month. The account offers a nominal annual interest rate of \( 3.6\% \), compounded monthly. He maintains this contribution schedule for a period of \( 10 \) years.
(a) Determine the total accumulated value of Myte’s savings plan at the completion of the \( 10 \)-year period.
Following the initial \( 10 \) years, Myte withdraws \( 100\,000 \text{ EUR} \) from the fund to provide a down payment for a property. He then transfers the remaining balance into a different high-yield account for a further \( 15 \) years. This new account earns a nominal annual interest rate of \( 4.5\% \), compounded quarterly.
(b) Calculate the final balance in Myte’s account at the end of this second \( 15 \)-year investment phase.
Most appropriate topic codes (IB Mathematics: applications and interpretation):
• SL 1.7: Calculation of annuities and the use of technology — part (a)
• SL 1.4: Compound interest problems with periodic compounding — part (b)
• SL 1.4: Compound interest problems with periodic compounding — part (b)
▶️ Answer/Explanation
(a)
This is an annuity with:
\( N = 10 \times 12 = 120 \) months
\( I\% = 3.6 \)
\( PV = 0 \)
\( PMT = -1500 \)
\( P/Y = 12 \), \( C/Y = 12 \)
Using financial solver:
FV ≈ \( 216278.58 \)
\( \boxed{216278.58 \text{ EUR}} \)
(b)
Amount after withdrawal:
\( 216278.58 – 100000 = 116278.58 \)
Now compound for 15 years at 4.5% quarterly:
\( N = 15 \times 4 = 60 \) quarters
\( I\% = 4.5 \)
\( PV = -116278.58 \)
\( PMT = 0 \)
\( P/Y = 4 \), \( C/Y = 4 \)
Using financial solver:
FV ≈ \( 227515.92 \)
\( \boxed{227515.92 \text{ EUR}} \)