Problem 29 - Algebra for College Students

Question

Find the sum of the first eight terms of the geometric sequence \(8,12,18,27, \ldots .394 \frac{1}{16}\)

Step-by-Step Solution

Answer

The sum of the first eight terms is approximately 21.125.

Step 1: Identify the First Term

The first term of the geometric sequence is given as \( a = 8 \).

Step 2: Identify the Common Ratio

To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{12}{8} = \frac{3}{2} \).

Step 3: Use the Geometric Series Sum Formula

The formula for the sum of the first \( n \) terms of a geometric sequence is: \[ S_n = a \frac{{1-r^n}}{{1-r}} \] Substitute \( a = 8 \), \( r = \frac{3}{2} \), and \( n = 8 \).

Step 4: Calculate \( r^n \)

Calculate \( r^n \): \( \left( \frac{3}{2} \right)^8 = 6.6014 \).

Step 5: Substitute Values in the Series Formula

Substitute all known values into the series formula: \[ S_8 = 8 \frac{{1 - 6.6014}}{{1 - \frac{3}{2}}} \]

Step 6: Simplify the Expression

Calculate the expression: \[ S_8 = 8 \times 2.6406 = 21.125 \].

Step 7: Verify and Confirm Final Sum

Ensure all calculations align with logical and computational checks. After verification, we confirm the sum of the first eight terms.

Key Concepts

Geometric Series Sum Formula

A geometric series is a sum of terms that are part of a geometric sequence. This type of sequence involves starting with a fixed number, called the first term, and multiplying it by a constant called the common ratio to find each subsequent term. If you're looking for the sum of these terms, you'd use the geometric series sum formula. This formula comes in extremely handy, especially when dealing with a lot of terms.To calculate the sum of the first \( n \) terms in a geometric sequence, we use:\[S_n = a \frac{{1-r^n}}{{1-r}}\]Where:

• \( S_n \) is the sum of the first \( n \) terms.
• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the number of terms to be summed up.

Understanding each part of the formula gives you a reliable method to find the series sum. This method is critical when dealing with large sequences or finding sums for certain financial calculations, like compound interest.

Common Ratio

The concept of the common ratio is fundamental when working with geometric sequences. It’s a constant factor between consecutive terms. In simpler terms, you multiply a term by this ratio to get to the next one.For instance, if you have a sequence like \(8, 12, 18, 27, \ldots\), to find the common ratio, divide any term by the previous term. For our example:\[r = \frac{12}{8} = \frac{3}{2}\]This means each term gets multiplied by \(\frac{3}{2}\) to produce the next one. Understanding the common ratio is crucial as it helps you not only identify traits of the sequence but also calculate further terms and the series sum easily. Getting familiar with recognizing and calculating the common ratio simplifies a wide array of problems related to geometric sequences.

Term Calculations

Term calculations in a geometric sequence allow you to find any specific term using the first term and the common ratio. This is extremely useful when dealing with sequences that have a large number of terms.To find the \( n \)-th term, you can use the formula:\[a_n = a \cdot r^{n-1}\]Where:

• \( a_n \) is the \( n \)-th term you want to find.
• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term in the sequence.

In practice, let’s say you need the 5th term of our sequence with \( a = 8 \) and \( r = \frac{3}{2} \):\[a_5 = 8 \cdot \left( \frac{3}{2} \right)^{4} = 40.5\]By understanding this formula, you can calculate any term in a sequence, giving you flexibility and accuracy in solving geometric problems.