Modelling with Sequences & Series (AQA A-Level Mathematics): Revision Notes

Examples of Modelling:

1. Population Growth (Geometric Sequence):

• Suppose a population of bacteria doubles every hour, and you start with 100 bacteria.
• The population after $\ n$ hours can be modelled by a geometric sequence: $P_n = 100 \times 2^{n}$
• If you want to know the total population after 6 hours: $P_6 = 100 \times 2^6 = 100 \times 64 = :highlight[6400 bacteria]$

2. Loan Repayments (Geometric Series):

• Consider a loan where you repay a fixed amount each month with interest. If you repay £500 monthly on a loan that charges 1% interest per month on the remaining balance:
• The balance after each payment decreases according to a geometric series.
• If the initial loan is £10,000 and monthly interest rate is 1%: $R_n = 10000 \times (1.01)^n - \frac{500 \times (1.01^n - 1)}{0.01}$
• This series helps to determine how many months it will take to pay off the loan.

3. Savings Plan (Arithmetic Sequence):

• Suppose you save £100 in the first month, and each subsequent month you increase your savings by £10.
• The amount saved after$\ n$ months forms an arithmetic sequence: $S_n = 100 + (n-1) \times 10 = 90 + 10n$
• The total savings after $\ n$months: $T_n = \frac{n}{2} \left(2 \times 100 + (n-1) \times 10 \right) = \frac{n}{2} \times (200 + 10n - 10) = \frac{n}{2} \times (10n + 190)$

Example Exam Question:

Question: A company decides to increase its annual bonus to employees by £500 each year. In the first year, the bonus is £2000. Find the total bonus an employee would receive over 10 years.

[4 marks]

Solution:

4. Identify the Sequence: The bonus amounts form an arithmetic sequence with $\ a_1 = :highlight[£2000]$and $\ d = :highlight[£500] .$

5. Use the Formula for the Sum of an Arithmetic Sequence: $S_n = \frac{n}{2} \times (2a_1 + (n-1)d)$ Here $\ n = :highlight[10] \, \ a_1 = :highlight[2000] \, and \ d = :highlight[500] .$

6. Substitute the Values: $S_{10} = \frac{10}{2} \times (2 \times 2000 + (10-1) \times 500)$ $= 5 \times (4000 + 4500) = 5 \times 8500 = :highlight[42500]$ Final Answer: The total bonus over 10 years is £42,500.