Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum. $$\sum_{n=0}^{6} 3^{n}$$

Step-by-Step Solution

Answer

The sum of the series is 1093

Step 1: Identify the Common Ratio, First Term, and Number of Terms

In the given series \( \sum_{n=0}^{6} 3^{n} \), we can identify the following: \n The common ratio \( r = 3 \), since each term is a power of 3 \n The first term \( a = 3^0 = 1 \), since n starts at 0 \n The number of terms \( n = 6 + 1 = 7 \), since n is inclusive of both ends and starts from 0

Step 2: Use the formula for the sum of geometric series

We use the formula for the sum of geometric series: \( S = \frac{a(r^n - 1)}{r-1} \) Substituting the values we have, we get \( S = \frac{1(3^7 - 1)}{3-1} \)

Step 3: Simplify the equation

Solving the equation, we get: \( S = \frac{(2187 - 1)}{2} = 1093\)

Key Concepts

Sum of Series

The concept of the "Sum of Series" is central to understanding sequences like geometric series. Summing up a series means finding the total quantity when all the terms in the series are added together. In the given exercise, the series notation \( \sum_{n=0}^{6} 3^{n} \) represents the sum of terms from \( 3^0 \) to \( 3^6 \).

The formula for the sum of a finite geometric series is given by:

• \( S = \frac{a(r^n - 1)}{r-1} \)

where:

• \( a \) is the first term
• \( r \) is the common ratio
• \( n \) is the number of terms

To find the sum of this series, we substitute the values into the formula and simplify it. The essence of the series' sum lies in factoring exponential growth, making it easier to compute potential infinite sequences when limited to specific bounds.

Applying the finite geometric series formula ensures the calculated sum reflects a neat, analytical solution instead of manual addition, especially when dealing with larger numbers.

Common Ratio

The "Common Ratio" is a key part of geometric series. It is the factor by which we multiply each term to get the next term. For the series \( \sum_{n=0}^{6} 3^{n} \), the common ratio \( r \) is \( 3 \). This is because each subsequent term is 3 times the previous term.

Identifying the common ratio helps in understanding how quickly a sequence progresses. For example, in the series starting with \( a = 3^0 = 1 \), the terms are:

• \( 1 \)
• \( 3 \)
• \( 9 \), and so on...

A common ratio greater than 1 results in a rapidly increasing series. Conversely, a common ratio between 0 and 1 would signify a decreasing series. Understanding the common ratio is crucial, as it dictates the behavior of the entire series.

Arithmetic Progression

Though the given series is a geometric one, comparing it to "Arithmetic Progression" can deepen understanding. In arithmetic progression (AP), each term increases by a constant difference. For example, the sequence 2, 4, 6, 8 has a common difference of 2.

Geometric series, like the one explored, differ because they multiply by a constant (the common ratio) instead of adding. If the series were AP, instead of multiplying by 3, you would add a fixed number each time. Recognizing these differences helps in applying the correct formulas.

Key differences to note:

• Arithmetic Progression (AP): difference between each term is constant.
• Geometric Series: ratio between each term is constant.

While AP and geometric series might seem similar initially, they play entirely different roles in mathematics modeling and applications.