Problem 25 - Algebra for College Students

Question

Find the sum of the first ten terms of the geometric sequence \(1,2,4,8, \ldots\) 1023

Step-by-Step Solution

Answer

The sum of the first ten terms is 1023.

Step 1: Identify Sequence Properties

This sequence is geometric. A geometric sequence follows the form where each term can be obtained by multiplying the previous term by a constant called the common ratio. Here, the first term \(a = 1\) and the common ratio \(r = 2\).

Step 2: Recall the Formula for the Sum of Geometric Series

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

Step 3: Plug Values into the Formula

Using the formula, plug in \(a = 1\), \(r = 2\), and \(n = 10\). This gives: \[S_{10} = 1 \frac{2^{10} - 1}{2 - 1}\].

Step 4: Calculate the Result

Evaluate the expression. First, calculate \(2^{10} = 1024\). Then subtract 1 to get 1023. Finally, divide by 1, which results in 1023.

Key Concepts

Sum of Geometric Series

A geometric series is a sum of terms that follow a geometric sequence. In simple terms, this means you add up a series of numbers where each number is multiplied by the same constant, known as the common ratio, to get the next term. To find the sum of this series, we use a special formula. This makes adding up all the numbers much easier, especially when there are many terms to consider.

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

• \( S_n = a \frac{r^n - 1}{r - 1} \)

Here, \( S_n \) is the sum of the series, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms you are summing up. This formula saves a lot of time because it helps you calculate the entire sum without having to add each term individually.

Let's take the example given: to find the sum of the first ten terms of the sequence \( 1, 2, 4, 8, \ldots\). By using the formula and plugging in \( a = 1 \), \( r = 2 \), and \( n = 10 \), the sum of these first ten terms is quickly found to be 1023. This illustrates how powerful the formula is when working with large sequences.

Common Ratio

The common ratio is a key feature of geometric sequences. It tells you how to get from one term to the next. In a geometric sequence, every term is obtained by multiplying the previous term by this common ratio.

For example, in the sequence \( 1, 2, 4, 8, \ldots \), the common ratio is 2. This means:

• \( 1 \times 2 = 2 \)
• \( 2 \times 2 = 4 \)
• \( 4 \times 2 = 8 \)

Recognizing the common ratio allows you to predict future terms easily and is also crucial when using the sum formula.

If you know the common ratio and the first term, you can easily identify any term in the sequence. It is also important in calculating the sum since it directly affects the geometric series formula \( S_n = a \frac{r^n - 1}{r - 1} \). The common ratio is the "r" in this formula, and understanding it helps you to solve many problems involving geometric sequences.

Geometric Sequence Formula

The geometric sequence formula helps you determine any term in a geometric sequence. Knowing how this works can simplify many problems, especially when dealing with large sequences.

The general formula to find the \( n \)-th term of a geometric sequence is

• \( a_n = a \cdot r^{n-1} \)

Here:

• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term position in the sequence.
• \( a_n \) is the \( n \)-th term you are looking for.

For example, if you know the first term is 1 and the common ratio is 2, to find the 10th term, you would use

• \( a_{10} = 1 \times 2^{10-1} = 1 \times 2^9 = 512 \).

This formula is very helpful as it lets you jump directly to any term without calculating all the preceding ones. It complements the sum formula, providing a full toolkit to tackle geometric sequences and series confidently.