Question

Find the sum of the series. $$ \sum_{n=1}^{10} 5\left(2^{n}\right) $$

Short answer

Answer: The sum of the finite geometric series is 10230.

Step-by-step solution

Identify the First Term, Common Ratio, and Number of Terms

We are given the series ∑(5 * 2^n) for n=1 to n=10. To find the sum, first, we need to identify key components of the series: - First term (a): Since n starts from 1, the first term in the series is when n=1. So, a = 5 * (2^1) = 5 * 2 = 10. - Common ratio (r): Looking at the series, we can observe that with each term, we multiply the previous term by 2. Therefore, the common ratio is r = 2. - Number of terms (n): The series goes from n=1 to n=10, so there are 10 terms. Now, we have all the necessary components to find the sum of the series.

Apply the Geometric Sum Formula

For a finite geometric series with the first term 'a', common ratio 'r', and 'n' terms, the sum is given by the formula: $$ S_n = a\frac{1-r^n}{1-r} $$ Now, we plug in the values of a, r, and n into the formula: $$ S_{10} = 10 \cdot \frac{1-2^{10}}{1-2} $$

Calculate the Sum

Perform the calculations within the parentheses before moving on: $$ S_{10} = 10 \cdot \frac{1-(1024)}{-1} $$ $$ S_{10} = 10 \cdot \frac{-1023}{-1} $$ $$ S_{10} = 10 \cdot 1023 $$ Finally, multiply the terms to get the sum: $$ S_{10} = 10230 $$ The sum of the given series is 10230.

Key concepts

Common Ratio

In a geometric series, the common ratio is a crucial component that defines how each term relates to its previous term. It is simply the factor by which we multiply one term to get the next.

In our given series, the common ratio is 2. This is because each successive term is obtained by multiplying the previous term by 2. For example, if we start with one of the terms as 10, the next term will be 10 multiplied by 2, resulting in 20. Then, 20 multiplied by 2 gives 40, continuing in this pattern.

A consistent common ratio across all terms confirms that it is indeed a geometric series, which helps us use the geometric series sum formula effectively.

First Term

The first term of a geometric series is represented by 'a', and it marks the starting point of the series. Knowing the first term is essential in determining the sum of the series. In our example, the series is given by \(\sum_{n=1}^{10} 5(2^n)\). Let's calculate the first term.

Since the series starts at n=1, substitute n=1 to find the first term:
* First term \(a = 5 \times (2^1) = 10\).

The progression of the series heavily relies on this initial value. Thus, ensuring we have the correct first term is critical when applying the geometric sum formula.

Sum Formula

The sum formula for a finite geometric series allows us to calculate the sum of all terms in the series quickly. It is formulated as follows:
\[S_n = a \frac{1-r^n}{1-r}\] where:

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

Upon applying this formula to our series, where \(a = 10\), \(r = 2\), and \(n = 10\), we compute:
\[S_{10} = 10 \times \frac{1-2^{10}}{1-2} \]
This simplifies to:
\[10 \times \frac{1-1024}{-1} \]
\[= 10 \times 1023 = 10230 \]
The sum of the first ten terms of this geometric series is 10230. The formula provides a streamlined way to sum the series without calculating each term individually.