Problem 22 - Basic Engineering Mathematics

Question

Find the sum of the first 9 terms of the series \(72.0,57.6,46.08, \ldots\)

Step-by-Step Solution

Answer

The sum of the first 9 terms is approximately 311.68.

Step 1: Identify the Sequence Type

The given series is 72.0, 57.6, 46.08, ... Since each term is a fraction of the previous term, this series is geometric.

Step 2: Determine the Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \( r = \frac{57.6}{72.0} = 0.8 \). Thus, the common ratio \(r\) is 0.8.

Step 3: Write the Formula for the Sum of a Geometric Series

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

Step 4: Assign Values for Variables

Assign the given values to variables: \(a = 72.0\), \(r = 0.8\), and \(n = 9\). Substitute these values into the sum formula.

Step 5: Calculate the Sum of the First 9 Terms

Substitute the values into the formula:\[ S_9 = 72.0 \frac{1 - (0.8)^9}{1 - 0.8} \]Calculate \((0.8)^9\):\((0.8)^9 \approx 0.134217728\)Next, compute:\[ S_9 = 72.0 \frac{1 - 0.134217728}{0.2} \approx 72.0 \times 4.32891136 \approx 311.68 \]Therefore, the sum of the first 9 terms is approximately 311.68.

Key Concepts

Common Ratio

In the world of sequences, especially geometric sequences, the common ratio is a vital concept. It tells you how much each term in the sequence is multiplied by to get the next term.

To find the common ratio, take any term in the sequence and divide it by the previous term. So, if you have a sequence like 72.0, 57.6, 46.08, ..., the common ratio \( r \) is calculated by dividing the second term by the first.

• Example calculation: \( r = \frac{57.6}{72.0} = 0.8 \)

This means each term is 0.8 times the term before it. Understanding this ratio is pivotal because it allows us to predict what the subsequent terms will be or reverse calculate the previous ones.

Whenever you encounter a sequence and each term seems to shrink or grow by the same scale factor, you've most likely stumbled upon a geometric series with a common ratio. This ratio, being consistent across the sequence, helps maintain the geometric nature of the sequence.

Sum of Geometric Series

When you want to know the total sum of the terms in a geometric sequence, you use a special formula called the sum of a geometric series. This formula is perfect for quickly adding up several terms without needing to calculate each one individually.

The formula for the sum \( S_n \) of the first \( n \) terms in a geometric series is:
\[ S_n = a \frac{1 - r^n}{1 - r} \]

• Where \( a \) is the first term,
• \( r \) is the common ratio,
• and \( n \) is the number of terms you want to sum.

By plugging in these values, you turn a potentially tedious process into something manageable.

Consider: if you know the first term is 72.0, the common ratio is 0.8, and you want the sum of the first 9 terms, you would compute:
\[ S_9 = 72.0 \frac{1 - (0.8)^9}{1 - 0.8} \]
Here, keeping track of exponents and subtracting precisely helps calculate efficiently. This formula not only saves time but ensures accuracy across the board.

Sequence Type Identification

Recognizing whether a series is arithmetic or geometric is the first key step in solving many sequence-related problems. Sequence type identification helps you select the right set of tools and formulas to use.

An arithmetic sequence increases by adding the same amount to each term. On the other hand, a geometric sequence, like our initial series, multiplies each term by the same common ratio to get to the next term.

• For instance, in the series 72.0, 57.6, 46.08, ..., each term is a fraction of the previous one, which indicates geometric.

Once you identify that a series is geometric, you can proceed to use the sum formulas specifically designed for geometric series.

Top tip: look for multiplying or dividing relationships between terms to spot a geometric sequence! Identifying this pattern early simplifies your path towards solving more complex problems, such as finding sums or missing terms.