Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=1, a_{8}=128, n=10 $$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 1023.

Step 1: Understanding the Geometric Sequence

First, note that a geometric sequence is defined by a starting term \(a_1\) and a common ratio \(r\). We have the first term \(a_1 = 1\) and the eighth term \(a_8 = 128\). We know the formula for the \(n\)-th term of a geometric sequence is given by \( a_n = a_1 \, r^{n-1} \).

Step 2: Finding the Common Ratio

Use the formula \( a_n = a_1 \, r^{n-1} \) to find the common ratio \(r\). Knowing that \( a_8 = 128 \), we have \( 128 = 1 \, r^{7} \). Solve for \(r\) by taking the seventh root: \( r = 128^{1/7} = 2 \).

Step 3: Using the Sum Formula for Geometric Series

The sum \(S_n\) of the first \(n\) terms in a geometric series is given by the formula \( S_n = a_1 \frac{r^n - 1}{r - 1} \). Here, \( a_1 = 1, r = 2, \) and \( n = 10 \). Plug these values into the sum formula.

Step 4: Calculating the Sum

Substitute the known values into the formula: \( S_{10} = 1 \frac{2^{10} - 1}{2 - 1} = 2^{10} - 1 \). Calculate \(2^{10} = 1024 \), so \(S_{10} = 1024 - 1 = 1023\).

Key Concepts

Sum of n terms

In a geometric series, finding the sum of a specific number of terms is important, as it allows you to understand the behavior of the series across this defined range. The formula for the sum of the first \(n\) terms is: \[ S_n = a_1 \frac{r^n - 1}{r - 1} \] where \(S_n\) represents the sum of the series, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms you're summing.

• Determine the first term, \(a_1\).
• Identify the common ratio, \(r\).
• Decide how many terms, \(n\), you want to sum.

By using these components, you can quickly calculate the sum of terms without having to add each individual term manually. For example, for our problem, the sum of the first 10 terms is calculated by substituting into the formula, resulting in \(S_{10} = 1023\). This is a practical and efficient way to handle series.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a non-zero constant called the common ratio. For example, consider the sequence: 1, 2, 4, 8, 16, etc.

• The first term \(a_1\) is the starting point of the sequence.
• Each subsequent term is calculated by multiplying the previous term by the common ratio, \(r\).
• A geometric sequence increases or decreases based on whether \(r > 1\) or \(0 < r < 1\), respectively.

This type of sequence is predictable and systematic, which makes it easy to analyze and use in various mathematical contexts. Understanding the structure of a geometric sequence is crucial for solving related problems, like finding specific terms or sums within the series.

Common Ratio

In a geometric sequence, the common ratio \(r\) is a key element that defines the pattern of the sequence. It is the factor by which each term is multiplied to obtain the following term. For instance, if you have the sequence 1, 2, 4, where each term is double the previous, the common ratio is 2. To find the common ratio:

• Use two consecutive terms from the sequence.
• Divide the second term by the first.
• For example, in the sequence with a first term of 1 and an eighth term of 128, the common ratio can be calculated as the seventh root of 128, resulting in \(r = 2\).

The correct identification of \(r\) is critical for both term calculation and sum determination.

Sum Formula

The sum formula for a geometric series provides a way to calculate the sum of a specified number of terms quickly. The formula \[ S_n = a_1 \frac{r^n - 1}{r - 1} \] allows you to find the sum efficiently without the need to add each term separately. Here’s why this formula is useful:

• It eliminates the repetitive task of adding each term individually.
• Makes the calculation of large series feasible and quick.
• Perfect for sequences with a large number of terms, as seen in the example problem.

By plugging in the known values, \(a_1 = 1\), \(r = 2\), and \(n = 10\), we simplify the process, resulting in a sum of 1023. Understanding and applying the sum formula is fundamental to mastering geometric series.