**Finance**

**Compound Interest**

In the realm of finance, geometric sequences find a pivotal role, particularly in the computation of compound interest. The future value A in an account, given a principal amount P, an interest rate r, and a number of periods n, is calculated using:

A = P(1 + r)^{n}

This formula is pivotal in financial planning and investment strategies, enabling individuals and corporations to forecast the future value of their investments and thereby make informed decisions.

- Example Question: If you invest £1000 at an interest rate of 5% per annum, how much will you have in the account after 10 years?Solution: Substituting P = £1000, r = 0.05, and n = 10 into the formula:A = 1000(1 + 0.05)
^{10}A = 1000(1.05)^{10}A = 1000(1.62889) A = £1628.89

Understanding the nuances of compound interest, such as the impact of the interest rate and the period, is crucial for financial literacy and effective monetary management.

**Loan Repayments**

Arithmetic sequences, on the other hand, find utility in modelling loan repayments. If a loan is being repaid in equal instalments, the outstanding balance forms an arithmetic sequence, which can be utilized to predict future balances and plan financial activities accordingly.

- Example Question: If a £5000 loan is being repaid over 5 years in equal annual instalments, what will be the outstanding balance after 3 years?Solution: If the annual repayment is x, after the first year, the outstanding balance will be £5000 - x, after the second year it will be £5000 - 2x, and so on. If we assume x to be £1000, after 3 years, the outstanding balance will be £5000 - 3x = £5000 - 3*£1000 = £2000.

Understanding the arithmetic sequence of loan repayments assists in financial planning and ensures adherence to repayment schedules.

**Physics**

**Radioactive Decay**

In physics, geometric sequences are employed to model radioactive decay. Given a substance with a half-life (time taken for half the substance to decay) of T years, the amount A of the substance left after n years is given by:

A = A0(1/2)^{(n/T)}

- Example Question: If a substance has a half-life of 3 years, how much of a 100g sample will remain after 9 years?Solution: Here, A0 = 100g, T = 3 years, and n = 9 years.A = 100(1/2)
^{(9/3)}A = 100(1/2)^{3}A = 100/8 A = 12.5g

Understanding radioactive decay through geometric sequences provides insights into the behaviour of radioactive substances over time, which is pivotal in fields like nuclear physics and medicine.

**Projectile Motion**

Arithmetic sequences can also be applied to calculate the position of an object in projectile motion at equal time intervals, given its initial velocity and launch angle. This application is crucial in various fields of physics, including mechanics and astrophysics, providing a mathematical framework to predict and analyse projectile motion.

**Population Growth**

**Exponential Growth**

In biology, if a population of organisms doubles every period, this can be modelled using a geometric sequence. The population P after n periods is given by:

P = P0(2)^{n}

- Example Question: If a bacteria colony doubles every hour, how large will a colony of 500 bacteria be after 5 hours?Solution: Here, P0 = 500 and n = 5.P = 500(2)
^{5}P = 500(32) P = 16000

**Logistic Growth**

In certain scenarios, population growth is not unlimited and may be restricted by factors like food availability. This can be modelled using more complex sequences and series, which consider the carrying capacity of the environment.

## FAQ

The nth term formula of a geometric sequence, a_{n} = a * r^{(n-1)}, inherently involves the concept of powers and exponents. Here, 'a' is the first term, 'r' is the common ratio (a non-zero number), and 'n' is the term number. The exponent (n-1) indicates that the common ratio is multiplied by itself (n-1) times when calculating the nth term. This relationship between geometric sequences and exponents provides a practical application of exponentiation in real-world scenarios, such as calculating compound interest, population growth, and more, by extending a constant rate of growth across multiple periods.

In computing, geometric sequences are often encountered in algorithms, particularly in those that involve divide-and-conquer strategies. An example is the binary search algorithm, which repeatedly divides the search space in half until the desired value is found. The running time of such algorithms is often logarithmic, and the sequence of problem sizes forms a geometric sequence. For instance, if an array of length n is repeatedly divided in half, the sizes of the arrays in each step of the algorithm form a geometric sequence: n, n/2, n/4, n/8, ..., 1.

Yes, geometric sequences can be used to model and analyse the spread of diseases in epidemiology, particularly in the early stages of an outbreak where exponential growth is often observed. The basic reproduction number, R0, which represents the average number of secondary infections produced by one infected individual in a completely susceptible population, forms the common ratio in this geometric sequence. If one individual initially is infected, and they infect R0 others, and each of those infects R0 others, the total number of cases after n generations of infection can be modelled as a geometric sequence: 1, R0, R0^{2}, R0^{3}, ..., R0^{(n-1)}. This model simplifies the complex dynamics of disease spread but provides a useful approximation in the early phases of an outbreak.

Geometric sequences are widely used in biology to predict population growth under the assumption that the population grows at a constant rate. This is often referred to as exponential growth. The formula to calculate future population size (N) is N = N0 * e^{(rt)}, where N0 is the initial population size, e is the base of the natural logarithm (approximately equal to 2.71828), r is the per capita growth rate, and t is time. This model assumes that resources are unlimited and that populations can grow indefinitely, which is an idealization as in real-world scenarios, environmental resistance usually slows growth as populations increase.

In finance, particularly in the calculation of compound interest, geometric sequences play a pivotal role. The formula to calculate the future value (A) of an investment compounded annually is A = P(1 + r/n)^{(nt)}, where P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per unit t, and t is the time the money is invested for, in years. This formula is derived based on a geometric sequence where each term represents the amount for each period and is found by multiplying the previous one by a fixed, non-zero number - in this case, (1 + r/n).

## Practice Questions

The question involves a geometric sequence since the bacteria population doubles (a common ratio of 2) every 3 hours. The formula for the nth term of a geometric sequence is given by: a_{n} = a * r^{(n-1)} where:

- a is the first term,
- r is the common ratio,
- n is the term number.

Given that a = 200, r = 2, and since the bacteria doubles every 3 hours, after 24 hours, n = 24/3 = 8.

a_{8} = 200 * 2^{(8-1)} a_{8} = 200 * 2^{7} a_{8} = 200 * 128 a_{8} = 25600

Thus, there will be 25,600 bacteria present after 24 hours.

The outstanding balance after each payment forms an arithmetic sequence. If the annual repayment is x, after the first year, the outstanding balance will be £12,000 - x, after the second year it will be £12,000 - 2x, and so on until after the fifth year it will be £12,000 - 5x.

Since the loan is fully repaid after 5 years, we have: 12,000 - 5x = 0 5x = 12,000 x = 12,000/5 x = 2,400

Therefore, the annual repayment to repay the loan over 5 years is £2,400.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.