Understanding the Sum of a Geometric Series with a Practical Example

When dealing with sequences in mathematics, one of the most common types is the geometric series. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this article, we will explore the formula for the sum of a geometric series and apply it to a practical problem involving a series of the form an where the base is 2.

The formula for the sum of the first n terms of a geometric series is S a cdot frac{r^n - 1}{r - 1}, where a is the first term, r is the common ratio, and n is the number of terms.

Deriving the Sum Formula of a Geometric Series

The sum of a geometric series can be derived through a simple algebraic manipulation. Starting with the sum S of the series:

S a ar ar^2 ar^3 cdots ar^{n-1}

When we multiply the entire sum by the common ratio r, we get:

rS ar ar^2 ar^3 ar^4 cdots ar^n

Now, subtract the original sum from this new equation:

rS - S (ar ar^2 ar^3 cdots ar^n) - (a ar ar^2 ar^3 cdots ar^{n-1})

This simplifies to:

rS - S -a ar^n

Factoring out the common term S on the left side gives:

S(r - 1) a(r^n - 1)

Dividing both sides by (r - 1) results in the sum formula:

S a cdot frac{r^n - 1}{r - 1}

Examples and Practical Applications

Let's apply this formula to a specific example: the sum of the series where each term is 2 raised to the power of the term's index. This can be written as:

S 2 2^2 2^3 2^4 cdots 2^n

Observing this series, we can see that we can factor out a 2 from every term after the first, which gives us:

S 2 2 cdot 2 2 cdot 2^2 2 cdot 2^3 cdots 2 cdot 2^{n-1}

Here, it is clear that a 2, r 2, and n n. Plugging these values into the sum formula:

S 2 cdot frac{2^n - 1}{2 - 1} 2^{n-1} - 2

Conclusion

The sum of a geometric series can be calculated using the formula S a cdot frac{r^n - 1}{r - 1}. This article has demonstrated the derivation of this formula and its application to a specific geometric series. The common ratio, often denoted r, is a crucial factor in determining the series' behavior, and it affects the growth of the series significantly. Understanding this concept is essential in various fields, from finance to computer science, where geometric series and their properties are often encountered.

Key Takeaways

Geometric series involve multiplying each term by a fixed non-zero number, known as the common ratio. The sum formula for a geometric series is S a cdot frac{r^n - 1}{r - 1}. For a series where each term is a power of a base (such as 2), the sum formula can be simplified by factoring out the base.