Problem 28 - Algebra for College Students

Question

Find the sum of the first ten terms of the geometric sequence \(5,10,20,40, \ldots .5115\)

Step-by-Step Solution

Answer

The sum of the first ten terms is 5115.

Step 1: Identify the elements of the sequence

The first term of the geometric sequence is given as \( a = 5 \). To find the common ratio \( r \), we divide the second term by the first term: \( r = \frac{10}{5} = 2 \). Therefore, the elements are: first term \( a = 5 \) and common ratio \( r = 2 \).

Step 2: Formula for the sum of a geometric sequence

The formula to find the sum of the first \( n \) terms of a geometric sequence is given by: \[ S_n = a \frac{r^n - 1}{r - 1} \] where \( S_n \) is the sum, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 3: Calculate the sum of the first 10 terms

Substitute \( a = 5 \), \( r = 2 \), and \( n = 10 \) into the sum formula. \[ S_{10} = 5 \frac{2^{10} - 1}{2 - 1} \] This simplifies to: \[ S_{10} = 5 (2^{10} - 1) \] Calculate \( 2^{10} = 1024 \). Then, \[ S_{10} = 5 (1024 - 1) = 5 \times 1023 = 5115 \]

Key Concepts

Sum of Geometric Sequence

The sum of a geometric sequence is a crucial concept when dealing with series. A geometric sequence is a set of numbers where each term is obtained by multiplying the previous one by a constant called the common ratio. Understanding how to find the sum can be very useful, especially in mathematical and financial calculations.
The formula for finding the sum of the first \( n \) terms of a geometric sequence is: \[ S_n = a \frac{r^n - 1}{r - 1} \] Here, \( S_n \) is the sum of the terms you want to find, \( a \) is the first term, \( r \) is the common ratio, and \( n \) represents the number of terms.
To effectively use this formula, you need to know the first term, the common ratio, and how many terms you are summing. Let's take a closer look at the other individual components that contribute to understanding and using this formula effectively.

Common Ratio

The common ratio in a geometric sequence is the factor by which each term is multiplied to yield the next term. It plays a central role in defining the sequence. To find the common ratio, simply divide any term in the sequence by the previous term.
For the given sequence:

• Second term: 10
• First term: 5
• Common ratio \( r = \frac{10}{5} = 2 \)

This means that each term is double the previous term.
Recognizing the common ratio allows you to predict the structure of the sequence and apply the sum formula correctly. It dictates how quickly the terms grow as you move along the sequence. If \( r = 1 \), the sequence is constant, while for \( r > 1 \), the sequence grows exponentially.

First Term of Sequence

The first term of a geometric sequence, denoted \( a \), is foundational in calculating the sum of a sequence. It's the starting point and anchor for the entire series of terms. For our example, the first term \( a \) is 5.
This initial element is crucial since every other term is derived by multiplying this starting term by the power of the common ratio (r) for the respective term position \( n-1 \).
Accurate determination of the first term ensures precise calculation for the subsequent terms, and ultimately, the sum of the sequence. In practical applications, knowing \( a \) enables you to apply the formula for the sum of the geometric sequence with confidence, as it directly influences the cumulative value of the terms you're adding together.