1.3 Geometric sequences and series questions for IB Maths AI

Write down an expression for the salary in year $n$ for

(i) Job A;

(ii) Job B.

Find the first year in which the salary for Job B is greater than the salary for Job A.

State one reason why Leilani might still choose Job A.

Calculate the national debt per person at the start of 2020. Give your answer to the nearest dollar.

An economist models the national debt at the start of each year as a geometric sequence. The debt increases by 7.1% each year.

Write down the common ratio of the sequence.

Use the model to find the national debt at the start of 2030. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

The population of Ardonia at the start of 2030 is predicted to be $3.44\times {10}^8$.

Find the predicted national debt per person at the start of 2030. Give your answer to the nearest dollar.

Find the first year in which, according to the model, the national debt at the start of the year is greater than $1.00\times {10}^{13}$ USD.

Give one reason why this model may not be suitable for predicting the national debt in 2060.

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 $u_2$ and $u_3$, giving your answers in the form $a+b\mathrm{i}$.

Show that $\mathrm{\mid }{u}_1\mathrm{\mid }$, $\mathrm{\mid }{u}_2\mathrm{\mid }$, $\mathrm{\mid }{u}_3\mathrm{\mid }$ are the first three terms of a geometric sequence with common ratio $0.5$.

Find the smallest value of $n$ for which $\mathrm{\mid }{u}_n\mathrm{\mid }<0.01$.

Let $u_1$ be the number of seats in row 1 and $d$ the number of extra seats in each row. Write down two equations in $u_1$ and $d$.

Find the value of $u_1$ and the value of $d$.

For a concert, at least 2500 seats are needed. The seats are used from row 1 backwards. Find the least number of rows that must be used.

For this concert, a seat in row 1 costs 85 euros (EUR). The price in each row is 5% less than the price in the row in front of it. Find the price of a seat in row 12. Give your answer correct to two decimal places.

Find how many rows have a seat price of at least EUR 50.

The organiser says that a seat in row 30 costs less than a quarter of the price of a seat in row 1. Determine whether the organiser is correct.

Write down the matrix $\mathit{L}$ such that $\left(\begin{array}{c}{J}_{n+1}\\ A_{n+1}\end{array}\right)=\mathit{L}\left(\begin{array}{c}{J}_n\\ A_n\end{array}\right)$.

(i) Find $J_1$ and $A_1$.

(ii) Find the total number of females in the reserve after 6 years, according to this model.

Show that the eigenvalues of $\mathit{L}$ are $1.1$ and $-0.3$.

Find an eigenvector of $\mathit{L}$ for each eigenvalue.

Show that $\left(\begin{array}{c}310\\ 130\end{array}\right)=50\left(\begin{array}{c}5\\ 3\end{array}\right)+20\left(\begin{array}{c}3\\ -1\end{array}\right)$.

Hence show that the total number of females after $n$ years is $T_n=400(1.1{)}^n+40(-0.3{)}^n$.

Find the first year in which the total number of females is predicted to exceed 3000.

Describe the long-term behaviour of the beaver population according to this model. Justify your answer.

The reserve cannot support an ever-growing population, so the ecologists plan to relocate some adult females each year. This changes only the proportion of adult females that remain as adults in the reserve, from 0.6 to $a$.

The population is stable in the long term when $\mathit{L}$ has an eigenvalue equal to 1.

Find the value of $a$ for which the population is stable.

The natural survival rate of adults remains 60 %, and the relocated females are chosen from the adults who would otherwise have remained.

Find the percentage of these adult females that must be relocated each year.

Explain what the model predicts if the ecologists relocate slightly more than this percentage every year.

State two limitations of this model.