2. Geometric Series

A geometric sequence is one in which each term is found by multiplying the previous term by a fixed number called the common ratio 𝑟.

A geometric series is the sum of terms in a geometric sequence.

A geometric series builds on a geometric sequence. In a geometric sequence, each term is calculated by multiplying the previous term by a constant value, known as the common ratio (r). For example, if the first term is 2 and the common ratio is 3, the sequence would be 2, 6, 18, 54, and so on. Each number is simply the last number multiplied by 3. A geometric series is then the addition of all these terms together. Understanding this concept is key to grasping how these sequences can represent real-life situations where growth is multiplicative, like population growth or interest calculations.

Imagine you start a savings account with $100, and you get 50$150 (100 + 50% of 100). At the end of the second year, you would have $225 (150 + 50% of 150), and so forth. Each year, your total grows by multiplying the previous total by 1.5 (which is your growth factor). The series of amounts you have at the end of each year is a geometric series.

Let the first term be 𝑎, and the common ratio be 𝑟. The geometric sequence looks like:

𝑎, 𝑎𝑟, 𝑎𝑟², 𝑎𝑟³,…, 𝑎𝑟𝑛−1

The sum of the first 𝑛 terms, 𝑆, of a geometric series is:

𝑆 = 𝑎(1−𝑟ⁿ) / (1−𝑟) for 𝑟 ≠ 1.

The general form of a geometric series provides a way to calculate the sum of the first 'n' terms easily. Here, 'a' is the first term of the sequence, and 'r' is the common ratio. The formula shows how the sum (S) can be calculated based on the first term and the growth rate. The term (1 - r^n) indicates how the series grows as n increases. This formula is valid as long as the common ratio (r) is not equal to 1 because, if r were 1, every term would be the same, leading to a simple sum of n times the first term.

Think of a plant that doubles its height every month. If it starts at 1 meter, after one month it is 2 meters (1 meter x 2), after two months it is 4 meters (2 meters x 2), and so on. If you want to find out how tall the plant will be after the first 5 months, you can use the geometric series formula with a = 1 meter, r = 2, and n = 5. The formula gives you a straightforward way to calculate the total growth of the plant over these months.

Find the sum of the first 5 terms of the geometric series:

2, 6, 18, 54,…

Solution: • First term 𝑎 = 2 • Common ratio 𝑟 = 3 • Number of terms 𝑛 = 5

𝑆 = 2(1−3⁵) / (1−3) = 2(1−243) / (−2) = 2(−242) / (−2) = 242.

In this example, we have a geometric series starting at 2, with a common ratio of 3 and 5 terms. By inserting these values into the formula, we first calculate r^n, which is 3^5, equaling 243. Then, we substitute this value back into the sum formula. After simplifying, we find that the sum of these series equals 242. This exercise highlights how to apply the formula in practical scenarios to derive sums quickly.

Suppose you have a special offer where the first gift costs $2, and every subsequent gift triples in value. If you want to calculate the total value of the first 5 gifts, you can apply the same reasoning as in the example. The gifts increase in value rapidly, demonstrating how geometric growth works, just like a savings account that compounds interest.

--