Finding the Sum of the First 20 Terms of a Geometric Sequence

Understanding how to find the sum of the first n terms in a geometric sequence is a fundamental concept in mathematics, particularly in geometry and algebra. This article will explore a specific problem involving the sum of the first 20 terms of the sequence 2, 4, 8, 16, illustrating the process step-by-step.

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the given sequence, the common ratio r is 2, as each term is twice the previous one.

Identifying the First Term and Common Ratio

The given sequence is 2, 4, 8, 16. The first term a is 2, and the common ratio r is 2. This can be confirmed by simple division of successive terms:

r 4/2 8/4 16/8 2

Sum of the First n Terms Formula

The sum of the first n terms of a geometric series with first term a and common ratio r is given by the formula:

Sn a frac14;1 - rnfrac14; / 1 - r

Applying the Formula to the Given Problem

For the given problem, we need to find the sum of the first 20 terms of the geometric sequence. We have:

First term, a 2 Common ratio, r 2 Number of terms, n 20

Substitute these values into the formula:

S20 2 frac14;1 - 220frac14; / 1 - 2

Step-by-Step Calculation

Let's break down the calculation step-by-step:

Calculate the numerator: 1 - 220 Calculate the denominator: 1 - 2 Divide the results from step 1 and step 2. Multiply the result by the first term (2).

Step 1: Calculate the numerator

1 - 220  1 - 1048576  -1048575

Step 2: Calculate the denominator

1 - 2  -1

Step 3: Divide the results from step 1 and step 2

-1048575 / -1  1048575

Step 4: Multiply the result by the first term (2)

2 * 1048575  2097150

Therefore, the sum of the first 20 terms of the given geometric sequence is 2097150.

Conclusion

Understanding and applying the formula for the sum of the first n terms of a geometric sequence is a valuable skill in mathematics. By following the steps outlined in this article, you can confidently solve similar problems involving geometric sequences. Practice with various examples will further solidify your understanding of these concepts.

Additional Resources and Keywords for SEO

To enhance your knowledge on the topic, consider exploring these related keywords:

Geometric sequence Sum of terms Common ratio

For further reading, you might also want to look into:

Geometric progression Mathematical sequences and series Summation formulas