Sum of the First Six Terms of a Geometric Progression: A Comprehensive Guide

Understanding the concept of a geometric progression (GP) and calculating the sum of its terms is a fundamental skill in algebra. This article will walk you through the process of finding the sum of the first six terms of a geometric progression with the first few terms given as 2, 6, and 18. We will explore the steps involved, the formulas, and the mathematical computations to ensure a comprehensive understanding.

Understanding Geometric Progression

A geometric progression 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. This common ratio, denoted by r, is the constant factor between consecutive terms. For example, in the sequence 2, 6, 18, each term is obtained by multiplying the previous term by 3.

Step-by-Step Calculation

Identify the Common Ratio: The common ratio of a GP can be found by dividing any term by the previous term. In this case, we can verify the common ratio by dividing 6 by 2 and 18 by 6. Both divisions yield a common ratio of 3. Write the First Term and Number of Terms: The first term a is 2, and we need to find the sum of the first 6 terms, so n 6. Use the Formula for the Sum of the First n Terms of a GP: The formula for the sum Sn of the first n terms of a GP is given by Sn a (1 - rn) / (1 - r), where a is the first term and r is the common ratio. Substitute the Values into the Formula: Substituting a 2, r 3, and n 6 into the formula, we get:

S6 2 (1 - 36) / (1 - 3)

Computation and Verification

Calculate 36: 36 729. Substitute 36 into the Formula: S6 2 (1 - 729) / (1 - 3) Perform the Calculations: S6 2 (-728) / -2 2 times; 364 728.

Therefore, the sum of the first six terms of the geometric progression is 728.

Comprehensive Verification

Let's verify the calculation by listing the first six terms of the sequence: 2, 6, 18, 54, 162, 486. Adding these terms together, we get:

2 6 18 54 162 486 728

Alternative Formula and Explanation

Another way to express the sum of the first n terms of a GP is:

Sn a (rn - 1) / (r - 1)

In this case, for a 2, r 3, and n 6:

S6 2 (36 - 1) / (3 - 1)

Given that 36 729:

S6 2 (729 - 1) / 2 728

Conclusion

In conclusion, understanding and applying the formula for the sum of a geometric series enables us to determine the sum of the first six terms of the given GP, 2, 6, 18, to be 728. This method is not only efficient but also consistent with the principles of geometric progressions. By following these steps and using the appropriate formulas, you can solve similar problems with confidence.