Applications of series
Model growth, decay and financial problems using recurrence relations and geometric series; find limiting values of recurrence relations; apply compound interest and depreciation as geometric sequences.
Worked examples
Compound interest as a geometric sequence
Straightforward
Problem
is invested at per annum compound interest. Find the value after 6 years (to the nearest cent), and find how many complete years it takes to exceed .
1
Identify the geometric model for the balance after years.
Each year, the balance is multiplied by . So the balance after years is:
This is a geometric sequence with and .
2
Find the value after 6 years.
3
Find when the balance first exceeds .
Solve :
The investment first exceeds after complete years.
Answer
After 6 years the investment is worth , and it first exceeds after 6 complete years.
Depreciation and recurrence relations
Moderate
Problem
A machine is bought for and loses of its value each year. Set up a recurrence relation for its value after years, and find after how many complete years its value drops below .
1
Write the recurrence relation.
This gives .
2
Solve the inequality .
Since , dividing reverses the inequality:
Answer
The value first drops below after complete years.
Total salary as a geometric series
Challenging
Problem
A graduate starts on a salary of and receives a raise each year. Find the total salary paid over the first 6 years, to the nearest dollar.
1
Identify the geometric series.
Year 1: ; Year 2: ; ...; Year 6: . This is a GP with , , .
2
Calculate the sum using .
Compute :
Answer
The total salary paid over the first 6 years is approximately .
Practise
Q1·Straightforward
A sequence is defined by and . Find .
Explanation
This is a geometric sequence with and .
Q2·Straightforward
A sequence is defined by and . Find .
Explanation
Q3·Straightforward
$2000 is invested at per annum compound interest. Find the value of the investment after 3 years. Give your answer to the nearest cent.
Explanation
Q4·Straightforward
A car is purchased for $20\,000 and depreciates at per year. Find its value after 4 years, to the nearest dollar.
Explanation
Q5·Moderate
A salary starts at $60\,000 and increases by each year. Find the total salary earned over 5 years, to the nearest dollar.
Explanation
Salaries: $60\,000; $62\,400; $64\,896; $67\,492; $70\,191 (approx).
These form a GP with , :
These form a GP with , :
Q6·Moderate
A machine's value follows the recurrence relation with . Find to the nearest dollar.
Explanation
Alternatively: .
Q7·Moderate
The recurrence relation has a limiting value as . Find .
Explanation
Setting :
Q8·Moderate
A population of 50\,000 grows at per year. What will the population be after 10 years? Round to the nearest whole number.
Explanation
Q9·Moderate
$5000 is invested at per annum compound interest. In which year does the investment first exceed $8000? (Find the smallest whole number .)
Explanation
Using logarithms: .
So . Check: ; . ✓
Q10·Challenging
How many years does it take for an investment to double at per annum compound interest? Give your answer to 2 decimal places.
Explanation
Q11·Challenging
A machine is purchased for $50\,000 and depreciates at per year. After how many complete years is the machine worth less than $10\,000?
Explanation
Using logarithms:
Since , dividing reverses the inequality:
So the first complete year is .
Check: .
. ✓
Q12·Challenging
A superannuation fund receives a contribution of $5000 at the start of each year for 4 years, with interest compounding at per annum. Find the total value of the fund at the end of 4 years, to the nearest dollar.
Explanation
Each deposit earns interest for 4, 3, 2, and 1 year respectively:
This is a GP with and , :
This is a GP with and , :
Open Math
Applications of series
Algebra / Functions · MAV-12-03
Name:
Date:
Q1Straightforward
A sequence is defined by and . Find .
Q2Straightforward
A sequence is defined by and . Find .
Q3Straightforward
$2000 is invested at per annum compound interest. Find the value of the investment after 3 years. Give your answer to the nearest cent.
Q4Straightforward
A car is purchased for $20\,000 and depreciates at per year. Find its value after 4 years, to the nearest dollar.
Q5Moderate
A salary starts at $60\,000 and increases by each year. Find the total salary earned over 5 years, to the nearest dollar.
Q6Moderate
A machine's value follows the recurrence relation with . Find to the nearest dollar.
Q7Moderate
The recurrence relation has a limiting value as . Find .
Q8Moderate
A population of 50\,000 grows at per year. What will the population be after 10 years? Round to the nearest whole number.
Q9Moderate
$5000 is invested at per annum compound interest. In which year does the investment first exceed $8000? (Find the smallest whole number .)
Q10Challenging
How many years does it take for an investment to double at per annum compound interest? Give your answer to 2 decimal places.
Q11Challenging
A machine is purchased for $50\,000 and depreciates at per year. After how many complete years is the machine worth less than $10\,000?
Q12Challenging
A superannuation fund receives a contribution of $5000 at the start of each year for 4 years, with interest compounding at per annum. Find the total value of the fund at the end of 4 years, to the nearest dollar.
Worked solutions and answers at openmath.au/year-12/advanced/sequences-and-series/applications-of-series