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Chapter 11: Problem 25

Use the formula for \(S_{n}\) to find the sum of the first five terms of each geometric sequence. In Exercises 25 and 26, round to the nearest hundredth. See Example 5. $$a_{1}=8.423, r=2.859$$

Short Answer

Expert verified
The sum of the first 5 terms is approximately 1196.84.

Step by step solution

01

- Understand the Formula

The sum of the first n terms of a geometric sequence is given by the formula: \[ S_{n} = a_{1} \frac{r^{n} - 1}{r - 1} \] where \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
02

- Identify Given Values

Identify the values from the problem: First term \(a_{1} = 8.423\), Common ratio \(r = 2.859\), Number of terms \(n = 5\).
03

- Substitute the Values into the Formula

Substitute the given values into the geometric series formula: \[ S_{5} = 8.423 \frac{2.859^{5} - 1}{2.859 - 1} \].
04

- Compute \(r^{n}\)

Compute \(r^{5}\): \[ 2.859^{5} = 265.0632 \] (rounded to four decimal places).
05

- Substitute and Simplify

Substitute \(r^{5}\) back into the formula: \[ S_{5} = 8.423 \frac{265.0632 - 1}{2.859 - 1} \] This simplifies to: \[ S_{5} = 8.423 \frac{264.0632}{1.859} \].
06

- Perform the Division and Multiplication

Calculate the fraction first: \[ \frac{264.0632}{1.859} \ \approx 142.0554 \] Then multiply by \(a_{1}\): \[ S_{5} \approx 8.423 \times 142.0554 \].
07

- Find the Final Sum

Finally, multiply to find the sum: \[ S_{5} \approx 8.423 \times 142.0554 = 1196.8421 \] Rounding to the nearest hundredth, the sum \(S_{5}\) is approximately 1196.84.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Geometric Series
A geometric series is the sum of the terms of a geometric sequence. To find the sum of a geometric series, we can use a specific formula: \[ S_{n} = a_{1} \frac{r^{n} - 1}{r - 1} \] Here, \( S_{n} \) represents the sum of the first \( n \) terms, \( a_{1} \) is the first term of the series, \( r \) is the common ratio, and \( n \) is the number of terms you want to sum.
Let's break it down:
  • If you know the first term \( a_{1} \), the common ratio \( r \), and the number of terms \( n \), you can find the total sum of those terms using this formula.
  • The formula works because it accounts for the exponential nature of the growth in a geometric sequence.
Understanding how to use this formula can make problems involving geometric series much easier to solve.
Geometric Sequence Formula
A geometric sequence follows this general formula for the \( n \)-th term: \[ a_{n} = a_{1} \times r^{(n-1)} \] Here, \( a_{n} \) is the \( n \)-th term, \( a_{1} \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
In the problem given:
  • The first term \( a_{1} \) is 8.423.
  • The common ratio \( r \) is 2.859.
  • We need to find the sum of the first 5 terms, so \( n \) is 5.
When you have these values, you can explore the behavior of the sequence and understand how each term relates to the previous one. The geometric sequence formula helps in identifying terms and is crucial for calculating the sum of a series.
Computational Steps in Geometric Sequences
Let's go through the computational steps to solve geometric sequence problems:
  • Step 1: Identify the Values 鈥� Understand and extract the important values like \( a_{1} \), \( r \), and \( n \).
  • Step 2: Apply the Formula 鈥� Plug these values into the geometric series sum formula: \[ S_{n} = a_1 \frac{r^n - 1}{r - 1} \]
  • Step 3: Calculate \( r^n \) 鈥� Compute the value of \( r \) raised to the power \( n \). In the example, \( 2.859^5 = 265.0632 \) (rounded to four decimal places).
  • Step 4: Substitute 鈥� Insert the value of \( r^n \) back into the formula and simplify: \[ S_{5} = 8.423 \frac{265.0632 - 1}{2.859 - 1} \]
  • Step 5: Perform Arithmetic 鈥� Calculate the fraction and then multiply by \( a_{1} \): \[ \frac{264.0632}{1.859} \rightarrow 142.0554 \rightarrow 8.423 \times 142.0554 \rightarrow 1196.8421 \approx 1196.84 \]
Following these steps will guide you through solving and understanding geometric sequence problems effectively.

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