What is Arithmetic and Geometric Sequence?

There are 2 types of growth or decay (decay is just the opposite of growth – instead of progressing, the values are regressing).

Arithmetic sequence and geometric sequence (to avoid confusion, we will use the term sequence – a neutral term as it can mean either growth or decay).

Let’s break down arithmetic and geometric sequences in simple terms.

Arithmetic Sequence:

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference. We can represent an arithmetic sequence as:

an​=a1​+(n−1)⋅d

Where:

• an​ is the nth term of the sequence.
• a1​ is the first term of the sequence.
• d is the common difference between consecutive terms.
• n is the term number.

In simpler terms, in an arithmetic sequence, each term is found by adding the same value to the previous term.

For example, consider the sequence: 2, 5, 8, 11, 14, …

Here, a1​=2 (the first term) and d=3 (the common difference). So, the nth term an​ can be found using the formula above.

Geometric Sequence:

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number. This fixed number is called the common ratio. We can represent a geometric sequence as:

an​=a1​⋅r(n−1)

Where:

• an​ is the nth term of the sequence.
• a1​ is the first term of the sequence.
• r is the common ratio between consecutive terms.
• n is the term number.

In simpler terms, in a geometric sequence, each term is found by multiplying the previous term by the same value.

For example, consider the sequence: 3, 6, 12, 24, 48, …

Here, a1​=3 (the first term) and r=2 (the common ratio). So, the nth term an​ can be found using the formula above.

If a quantity is always increasing or decreasing, but the changes do not have the regularity of either arithmetic or geometry, it is said to be monotonic or continuous.  All sequences are monotonic but monotonic sequences need not be arithmetic or geometrical.

Sometimes geometric sequences is called exponential sequences. Though both are closely related, exponential sequences refer to a pattern where the growth itself increases over time.

Exponential sequences are represented by the equation:

N (t)=N₀​⋅e^kt

Where:

• N(t) is the quantity at time t,
• N₀​ is the initial quantity,
• e is the base of the natural logarithm (approximately equal to 2.71828),
• k is the growth rate, and
• t is time.

In simpler terms, the formula says that the quantity at any given time t is equal to the initial quantity (N₀​) multiplied by the constant e raised to the power of the growth rate (k) times time (t).

While both exponential growth and geometric growth involve rapid increases over time, geometric growth specifically refers to a type of exponential growth where each term in a sequence is obtained by multiplying the previous term by a constant ratio. Exponential growth, on the other hand, is a broader concept that encompasses various forms of growth where the rate of increase is proportional to the current value.

In the financial world, geometric and exponential sequences often appear in scenarios involving compound interest and investment growth. Here are some common examples:

1. Compound Interest: When interest is compounded at regular intervals, the amount of money in an account grows exponentially. Each compounding period, the interest is added to the principal, leading to a geometric sequence of account balances over time.

2. Investment Growth: Many investments, such as stocks, bonds, and mutual funds, can exhibit geometric growth over time. For example, if an investment grows by a certain percentage each year, the value of the investment at each subsequent year forms a geometric sequence.
3. Retirement Savings: Retirement savings accounts, such as EPF, often experience geometric growth as contributions are made regularly and accumulate over time, earning compound interest or returns.
4. Loan Amortization: In loan amortization schedules, such as mortgages or car loans, the remaining balance decreases over time as payments are made. The interest portion of the payment reduces as the principal decreases, resulting in a geometric decrease in the outstanding balance.
5. Population Growth in Financial Models: Some financial models may incorporate population growth rates to predict market trends or demand for products and services. Population growth can be modelled using geometric sequences when the growth rate remains constant.
6. Revenue Growth in Businesses: Businesses often aim for revenue growth over time. If a company’s revenue grows by a certain percentage each year, the revenue figures form a geometric sequence, which can be used for financial planning and forecasting.
7. Dividend Growth: Some dividend-paying stocks increase their dividends at a constant rate over time. The dividends paid by such stocks form a geometric sequence, reflecting the growth in income for shareholders.

Examples in the natural world are:

1. Bacterial Growth: Bacteria reproduce through binary fission, where one bacterium splits into two identical daughter cells. Under favorable conditions, bacteria can exhibit exponential growth as each generation doubles in population size. This rapid proliferation is a key feature of bacterial colonies in environments such as petri dishes or within the human body during infections.
2. Population Growth: In certain species, particularly those with short generation times and abundant resources, populations can undergo exponential growth. For example, some species of insects, rodents, and plants can experience exponential population growth when conditions are favourable, leading to rapid increases in population size over relatively short periods.
3. Algal Blooms: Algae populations in aquatic ecosystems can undergo exponential growth under optimal conditions of sunlight, nutrients, and temperature. When these conditions are met, algae reproduce rapidly, leading to algal blooms. These blooms can have significant ecological impacts, such as depleting oxygen levels in water bodies and disrupting aquatic ecosystems.
4. Compound Interest in Finance: Compound interest in finance results in exponential growth of investments over time. When interest is reinvested and earns additional interest, the investment grows at an accelerating rate. This concept underlies the power of long-term saving and investing, where even small contributions can compound into substantial wealth over time.
5. Viruses and Epidemics: The spread of infectious diseases, particularly viruses, can exhibit exponential growth in susceptible populations. Each infected individual can transmit the virus to multiple others, leading to a cascade of new infections. In the early stages of an epidemic, the number of cases can double rapidly, causing the outbreak to escalate quickly if left unchecked.
6. Cancer Growth: Cancerous tumours can undergo exponential growth as cancer cells divide and proliferate rapidly. In the absence of treatment, tumours can grow exponentially, invading surrounding tissues and organs. Understanding the exponential growth of cancer is crucial for developing effective treatments and managing the disease progression.
7. Weathering and Erosion: In geological processes, the physical weathering and erosion of rocks and landforms can exhibit exponential growth over long periods. For example, the gradual wearing down of mountains by processes such as freeze-thaw cycles, wind abrasion, and water erosion follows exponential patterns as these forces act repeatedly over time.