Problem 25 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The sum of the first five terms is approximately 957.61.

Step 1: Understand the Formula for Geometric Series

The sum of the first \( n \) terms of a geometric sequence can be calculated using 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.

Step 2: Identify Given Values

From the problem, we know:- \( a_1 = 8.423 \)- \( r = 2.859 \)- \( n = 5 \)

Step 3: Substitute Values into Formula

Substitute the known values into the geometric series sum formula: \[ S_5 = 8.423 \frac{2.859^5 - 1}{2.859 - 1} \]

Step 4: Calculate \( r^n \)

Calculate \( 2.859^5 \):\[ 2.859^5 \approx 212.3076 \]

Step 5: Compute the Denominator

Calculate the denominator \( r - 1 \):\[ 2.859 - 1 = 1.859 \]

Step 6: Calculate the Fraction in Formula

Calculate the fraction: \[ \frac{212.3076 - 1}{1.859} \approx \frac{211.3076}{1.859} \approx 113.6856 \]

Step 7: Compute the Sum \( S_5 \)

Finally, calculate \( S_5 \) by multiplying \( a_1 \) with the result from Step 6:\[ S_5 = 8.423 \times 113.6856 \approx 957.61 \]

Step 8: Round the Result

Round \( 957.61 \) to the nearest hundredth (it's already rounded):\[ S_5 = 957.61 \]

Key Concepts

Geometric Series Formula

The formula for finding the sum of a geometric series is a powerful tool. It helps us determine the total sum of several terms in a sequence where each term after the first is found by multiplying the previous one by a fixed number. This fixed number is known as the common ratio. The formula is written as:\[ S_n = a_1 \frac{r^n - 1}{r - 1} \]

• Where \( S_n \) represents the sum of the first \( n \) terms.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the total number of terms you want to sum up.

Understanding this formula is crucial because it lets you calculate the entire sum efficiently without needing to add each term individually. This saves time and effort, particularly in larger sequences.

Common Ratio

The common ratio is a fundamental part of a geometric sequence. It is the number that you multiply each term by to get to the next term. In this particular exercise, the common ratio \( r \) is given as 2.859. Knowing the common ratio helps you predict and calculate terms in the sequence.

• If \( r > 1 \), the sequence grows larger with each term.
• If \( 0 < r < 1 \), terms get smaller.
• If \( r < 0 \), terms will alternate in sign.

To compute \( r^n \), you simply raise the common ratio to the power of \( n \), which tells you how the sequence changes over \( n \) terms. For this exercise, you calculate \( 2.859^5 \) to see how the fifth term compares to the starting one.

Sum of Terms

Finding the sum of the terms in a geometric sequence requires careful calculations. Using the geometric series formula, we plug in our values:

• Calculate \( r^n \) which in this case is \( 2.859^5 \), equivalent to approximately 212.3076.
• The denominator \( r - 1 = 1.859 \) is calculated to use in the formula.
• Substitute these into the equation to find the fractional part: \[ \frac{212.3076 - 1}{1.859} \approx 113.6856 \]

Finally, multiply this result by the first term \( a_1 = 8.423 \) to get the sum for the first five terms, \( S_5 \), resulting in 957.61. This sum covers how much these terms add up to, which is essential for understanding the overall growth in the sequence.

Round to Nearest Hundredth

Rounding numbers in mathematical solutions ensures accuracy to the desired level of precision. When you round to the nearest hundredth, you're adjusting the number to two decimal places. This concept is simple yet crucial to prevent errors when reporting the sum of terms.For example, from our calculation, we found the sum \( S_5 \approx 957.61 \). To determine if this is accurate to the nearest hundredth,

• You observe the third decimal place, which influences whether to round up or down.
• In our case, the value is already at 957.61, precisely to two decimal places.

So, no further adjustment is needed. Rounding appropriately is important for many real-world applications where precision matters, such as in finance or scientific measurements.