1.4 Financial applications of geometric sequences questions for IB Maths AI

Find the value of her savings after 4 years. Give your answer correct to two decimal places.

Find the number of complete months it takes for the value of her savings to first exceed 8000 CAD.

Akira converts 800 000 JPY. Find the number of Australian dollars he receives.

Find an expression for $f^{-1}(x)$.

Find $f^{-1}(5000)$ and interpret your answer in context.

Akira deposits 8154.90 AUD in a savings account that pays a nominal annual interest rate of 3.8%, compounded quarterly. Find the value of his savings after 4 years. Give your answer correct to two decimal places.

The value of Akira's savings, in AUD, after $t$ years is $g(t)=8154.90{\left(1.0095\right)}^{4t}$.

At any time, Akira can convert Australian dollars back to yen at 1 AUD = 97.5 JPY, less a 1% commission, so that $h(x)=0.99\times 97.5x$ is the number of yen he receives for $x$ AUD.

Find $(h\circ g)(4)$, giving your answer to the nearest yen.

Find the number of complete years Akira must leave his money in the account before it can be converted into more than 1 000 000 JPY.

Find the monthly payment for Option A.

Find the total amount of interest Ines pays with Option A.

Find the amount Ines would still owe after 2 years with Option A.

Option B is advertised as a "flat-rate loan at only 4.2 % per year". For each year of the loan, interest equal to 4.2 % of the original amount borrowed is added to the debt, so the total debt after $n$ years, $u_n$, forms an arithmetic sequence. The total debt after 5 years is then repaid in 60 equal monthly payments.

Show that the total debt after 5 years is EUR 29 040.

Hence write down the monthly payment for Option B.

Find the nominal annual interest rate, compounded monthly, that Ines is actually being charged with Option B.

Ines says that Option B is cheaper than Option A because 4.2 % is less than 6.0 %. Explain why she is wrong.

Option C is a "growing payment" loan with the same interest rate as Option A, 6.0 % compounded monthly, which is 0.5 % per month. The first payment is EUR $P$, and each later payment is 1 % larger than the previous one, so that the payments form a geometric sequence.

Write down an expression, in terms of $P$ and $k$, for the $k$th payment.

The present value of a payment of $X$ made at the end of month $k$ is $\frac{X}{{1.005}^k}$.

Show that the sum of the present values of all 60 payments is

The loan is fully repaid when the sum of the present values of the payments equals the amount borrowed.

Hence find the value of $P$.

Find

(i) the size of the final payment;

(ii) the total amount Ines repays with Option C.

Option C costs more in total than Option A. Suggest one reason why Ines might still choose Option C.

Ines later estimates the value of the bakery as the sum of the present values of all its future monthly profits, again discounting at 0.5 % per month. She models the profit at the end of the first month as EUR 1500, with each month's profit 0.3 % greater than the previous month's, and assumes this continues indefinitely.

Find the value of the bakery according to this model.

Explain why this method of valuation could not be used if Ines assumed that the monthly profit grew by 0.6 % each month.

Find the amount Kenji must deposit now. Give your answer correct to the nearest yen.

A second bank offers the same nominal annual interest rate, but compounded yearly. Find how much more Kenji would need to deposit now with this second bank to reach the same amount in 6 years.

Find the amount he saves in month 18.

Find the total he has saved after 24 months.

Find the number of months it takes Haruto to save at least 750 000 JPY.

Haruto's sister, Yui, also wants to have 750 000 JPY. She invests 650 000 JPY in an account that pays a nominal annual interest rate of 3.1%, compounded monthly. She makes no further deposits.

Find the number of complete months until the value of Yui's investment first reaches 750 000 JPY.

Write down how many months earlier than Yui Haruto reaches 750 000 JPY.

Comment on whether Haruto's savings plan is realistic.

Find the value of the car after 5 years. Give your answer correct to the nearest euro.

Find the percentage of the original value that the car has lost after 5 years.

Marta plans to sell the car at the end of the first year in which its value is less than 10 000 EUR. Find the number of years after which she will sell the car.