Carrying Out the Plan

Applying the solution

Let’s execute the plan we made in the previous step.

We want to calculate the number of messages with 10 bulbs. Now, let’s apply the formula to calculate it.

$$\text{Number of messages with 10 bulbs} = 2^{10} = 1024 \text{ messages}$$

The real-world applications of the geometric sequence

Geometric sequences are prevalent in various real-life scenarios, such as:

• Compound interest: In finance, compound interest follows a geometric sequence. For instance, if we invest $100 at a 5% annual interest rate compounded yearly, the growth of our investment is akin to a geometric sequence. Each year, the amount grows by a fixed factor (1 + the interest rate). The sequence of amounts after each compounding period forms a geometric progression.

• Population growth: Population growth in certain scenarios can mirror a geometric sequence. If a population increases by a fixed percentage annually (say, 5%), the population size in successive years can form a geometric sequence. Each year’s population is a multiple of the previous year’s population by the growth rate.

• Bacterial growth: Similarly, the growth of bacterial colonies under ideal conditions can follow a geometric sequence. For example, if the number of bacteria doubles every hour, it mirrors a geometric progression. The number of bacteria at each hour forms a sequence by multiplying the previous term by a fixed factor.

These examples illustrate how geometric sequences are not just abstract mathematical concepts but are prevalent in various aspects of our daily lives, from finance to biological phenomena. It is important to observe that, in reality, such phenomena rarely exhibit perfect geometric progression in their growth; however, they closely adhere to the underlying pattern. They showcase consistent growth or decay over time, characterized by a fixed ratio between successive terms.

Geometric progressions are notable for their rapid growth or decline due to their consistent multiplication or division by a fixed ratio. In scenarios like compound interest, population growth, or bacterial proliferation, the sequence can grow or decline quite rapidly over time. This characteristic makes geometric progressions powerful and impactful in various real-life applications showcasing exponential growth or decay.