Problem 60 - Beginning and Intermediate Algebra

Question

Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$\sum_{i=1}^{8} 5(2)^{i}$$

Step-by-Step Solution

Answer

The sum of the terms of the geometric sequence is 1275, using the formula \(S_n = \frac{a(r^n - 1)}{r - 1}\), with values \(a = 5\), \(r = 2\), and \(n = 8\). So, \(\sum_{i=1}^{8} 5(2)^{i} = 1275\).

Step 1: Identify the values of 'a', 'r', and 'n'

In the given geometric sequence, the first term (a) is 5, which can be seen as the coefficient of the exponential term. The common ratio (r) is the base of the exponent, which is 2. The number of terms (n) is specified in the sum, which is 8.

Step 2: Use the formula for Sn

The formula for the sum of a geometric series (Sn) is: $$S_n = \frac{a(r^n - 1)}{r - 1}$$ Plug in the identified values of 'a', 'r', and 'n' into the formula: $$S_8 = \frac{5(2^8 - 1)}{2 - 1}$$

Step 3: Evaluate the expression

Calculate the value of the expression: $$S_8 = \frac{5(256 - 1)}{1} = \frac{5(255)}{1} = 1275$$

Step 4: State the answer

The sum of the terms of the given geometric sequence is 1275. So, the result is: $$\sum_{i=1}^{8} 5(2)^{i} = 1275$$

Key Concepts

Sum of a Geometric Series

A geometric series involves adding up the terms of a geometric sequence. Each term in a geometric sequence is produced by multiplying the previous term by a constant, known as the common ratio. To find the sum of a finite geometric series, we apply a specific formula designed for this purpose. This formula allows us to calculate the total sum quickly without needing to add each term individually.

The formula for the sum of the first \(n\) terms of a geometric series is:

• \(S_n = \frac{a(r^n - 1)}{r - 1}\)

Where:

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

By using this formula, you only need to know the first term, the common ratio, and the number of terms to find the sum of the entire series. This approach saves time and reduces the risk of error from manual addition of all terms.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It plays a crucial role in determining the characteristics of the sequence and series. Identifying the common ratio is essential when using the geometric series formula.

In a sequence such as \(5, 10, 20, 40,\ldots\), the common ratio is found by dividing any term by the term before it:

• For example, \(\frac{10}{5} = 2\), \(\frac{20}{10} = 2\), confirming the common ratio \(r = 2\).

Understanding and identifying the common ratio allows us to predict future terms in the sequence or solve related problems, such as calculating series sums. If the common ratio is greater than one, the series terms increase exponentially; if less than one, the terms decrease; and if negative, the terms alternate in sign.

Geometric Series Formula

The geometric series formula is an efficient tool for finding the sum of a geometric series. It provides a comprehensive way to sum up a series without having to manually calculate each term's contribution, greatly simplifying the process.

By plugging the values of the first term \(a\), the common ratio \(r\), and the number of terms \(n\) into the formula \(S_n = \frac{a(r^n - 1)}{r - 1}\), we are effectively computing the total of all terms in the series:

• This requires only basic operations such as exponentiation, multiplication, and subtraction, making it easy and accessible.

Example in Action

For the series \(5 \times (2)^1 + 5 \times (2)^2 + \ldots + 5 \times (2)^8\), identifying \(a = 5\), \(r = 2\), and \(n = 8\) allows us to plug these into our formula, yielding:

• \(S_8 = \frac{5(2^8 - 1)}{2 - 1} = 1275\)

Thus, the formula simplifies the problem-solving process, enabling us to derive the sum efficiently.