Understanding and Calculating the Sum of the First 5 Terms of a Geometric Sequence

This article aims to guide you through the process of understanding and calculating the sum of the first five terms of a geometric sequence, starting from an initial set of terms. Specifically, we will look at the sequence 7, 14, 28, and its continuation and the sum of its first five terms. Let's delve into the details.

The Given Sequence and Its Pattern

The provided sequence is: 7, 14, 28, 56, 112. Here, we can observe a clear pattern: each term is double the previous one. This is the characteristic of a geometric sequence.

Defining the Terms of a Geometric Sequence

A geometric sequence is defined by its first term a and a common ratio r. For the sequence 7, 14, 28, 56, 112:

The first term a 7. The common ratio r 14/7 2.

Using these values, we can generate the first five terms as follows:

a 7 a r 7 * 2 14 a r2 7 * 22 28 a r3 7 * 23 56 a r4 7 * 24 112

Sum of the First Five Terms

To find the sum of the first five terms, we use the sum formula for a geometric series:

Sn a * rn - 1 / (r - 1), where n is the number of terms.

In this case, n 5, a 7, and r 2. Substituting these values,

S5 7 * 25 - 1 / (2 - 1) 7 * 24 / 1 7 * 16 112

However, we need to consider the complete sequence up to the fifth term. Adding the first five terms explicitly:

7 14 28 56 112 217

This confirms that the sum of the first five terms is indeed 217.

Conclusion

The sequence 7, 14, 28, 56, 112 follows a pattern of doubling each term. The sum of the first five terms of this sequence is calculated using the properties of a geometric sequence and its formula. By understanding the common ratio and the sum formula, we can easily solve similar problems involving geometric sequences.

Further Reading and Resources

If you want to learn more about geometric sequences and series, consider exploring resources such as 'Geometric Progression - Wikipedia', which provides comprehensive information and real-world applications.