Understanding the Sum of Geometric Sequences: A Detailed Guide

In this article, we will explore the detailed process of finding the sum of geometric sequences, focusing on a specific example: the sum of the first 10 terms of the sequence 5, 10, 20, 40. We will also delve into the general formula for the sum of a geometric sequence, which is crucial for understanding and applying the concept in various scenarios.

Identifying the Pattern in the Sequence

The sequence provided is a geometric sequence, where each term is obtained by multiplying the previous term by a constant factor. For the sequence 5, 10, 20, 40, the first term (a1) is 5, and the common ratio (r) is 2. This means each subsequent term is 2 times the previous term. Let's list the first 10 terms of this sequence to visualize the pattern: 1st term: 5 2nd term: 10 (5 * 2) 3rd term: 20 (10 * 2) 4th term: 40 (20 * 2) 5th term: 80 (40 * 2) 6th term: 160 (80 * 2) 7th term: 320 (160 * 2) 8th term: 640 (320 * 2) 9th term: 1280 (640 * 2) 10th term: 2560 (1280 * 2)

Summing the Terms of the Sequence

To find the sum of the first 10 terms, we can add them individually. Here's a step-by-step breakdown: 5 10 15 15 20 35 35 40 75 75 80 155 155 160 315 315 320 635 635 640 1275 1275 1280 2555 2555 2560 5115 Thus, the sum of the first 10 terms of the sequence 5, 10, 20, 40 is 5115.

Using the Sum Formula for Geometric Sequences

There is a more efficient way to find the sum of a geometric sequence using the formula:

Sn a(r^n-1)/(r-1)

In this formula, a is the first term, r is the common ratio, and n is the number of terms. For our sequence (5, 10, 20, 40), a 5, r 2, and n 10. Substituting these values into the formula, we get:

Sn 5(2^10-1)/(2-1) 51024 - 1 51023

This simplifies to:

Sn 5115

Thus, the sum of the first 10 terms of the sequence 5, 10, 20, 40 is 5115.

Conclusion

In summary, we have explored how to find the sum of a geometric sequence by identifying the pattern, summing the terms individually, and using the sum formula. The sum of the first 10 terms of the sequence 5, 10, 20, 40 is 5115. Understanding and applying these concepts is crucial for solving problems involving geometric sequences and series. This knowledge can be applied in various fields, including finance, physics, and engineering.

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