Problem 51 - Algebra 2

Question

Find the sum of each geometric series. $$ \sum_{n=1}^{7} 144\left(-\frac{1}{2}\right)^{n-1} $$

Step-by-Step Solution

Answer

The sum of the geometric series is approximately 96.75.

Step 1: Identify the Components of the Geometric Series

In the given series, the first term is 144. The common ratio is \(-\frac{1}{2}\), and the number of terms is 7.

Step 2: Apply 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 3: Substitute Values into the Formula

Using the formula from Step 2, substitute \(a = 144\), \(r = -\frac{1}{2}\), and \(n = 7\).\[ S_7 = 144 \frac{1 - \left(-\frac{1}{2}\right)^7}{1 - \left(-\frac{1}{2}\right)} \]

Step 4: Calculate the Power \((-\frac{1}{2})^7\)

Calculate the power:\[(-\frac{1}{2})^7 = -\frac{1}{128}\]

Step 5: Simplify the Denominator

The denominator is \(1 - (-\frac{1}{2})\):\[1 + \frac{1}{2} = \frac{3}{2}\]

Step 6: Substitute the Calculated Values and Simplify

Substitute the values from Steps 4 and 5 into the formula:\[S_7 = 144 \frac{1 - \left(-\frac{1}{128}\right)}{\frac{3}{2}} = 144 \frac{1 + \frac{1}{128}}{\frac{3}{2}}\]

Step 7: Further Simplify the Expression

Simplify the fraction:\[1 + \frac{1}{128} = \frac{128}{128} + \frac{1}{128} = \frac{129}{128}\]Then compute:\[\frac{129}{128} \div \frac{3}{2} = \frac{129}{128} \times \frac{2}{3} = \frac{258}{384}\]

Step 8: Calculate the Final Sum

Finally, multiply by 144:\[S_7 = 144 \times \frac{258}{384} = 144 \times 0.671875 \]

Step 9: Final Calculation

Calculate the final result:\[ S_7 \approx 96.75 \]

Key Concepts

Sum of Geometric Series

The sum of a geometric series is an important concept in mathematics, especially when dealing with sequences and series. A geometric series is formed by multiplying each term by a constant factor, known as the common ratio, to achieve the next term.
To find the sum, we use a specific formula that considers the first term, the common ratio, and the number of terms, often denoted by \(n\). The formula for the sum \(S_n\) of the first \(n\) terms is: \[S_n = a \frac{1 - r^n}{1 - r}\] where:

• \(S_n\) is the sum of the series.
• \(a\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the total number of terms.

This formula only applies when the common ratio \(r\) is not equal to 1, as otherwise the series does not converge to a specific value. By understanding this fundamental concept, you are equipped to tackle any problem involving geometric series efficiently.

First Term

The first term in a geometric series plays a crucial role as it serves as the starting point for generating all subsequent terms.
In a geometric series, each term is produced by multiplying the previous term by the common ratio.
This makes the first term, often denoted as \(a\), an essential component in the series formation.

• In the context of the problem, the series starts with \(a = 144\).
• The value of \(a\) directly affects the calculated sum, as it is multiplied by the fraction involving the common ratio raised to the power of the number of terms.

It's the anchor of the entire series, and any changes to this term can significantly alter the entire sequence.

Common Ratio

The common ratio is a defining element of a geometric sequence. It dictates how the sequence progresses from one term to the next. Simply put, it's the factor by which each term is multiplied to reach the following term.
In the example given, the common ratio is \(-\frac{1}{2}\), meaning each term is half of the previous term, but in the opposite direction (negative).

• A constant common ratio implies that every two consecutive terms have a uniform proportional relationship.
• This consistency allows for notable patterns within the sequence, making it easier to predict future terms or sum them using the geometric progression formula.
• The common ratio also crucially impacts the series sum calculation since it is raised to the power \(n\) when using the sum formula.

Understanding the common ratio is essential for accurately analyzing and manipulating geometric sequences.

Geometric Progression Formula

The geometric progression formula is the mathematical expression used to calculate the sum of terms in a geometric sequence. Each term of the sequence grows by multiplying the preceding term by a constant ratio. As such, the geometric progression formula helps determine the sum of a finite sequence quickly and efficiently.

• For a geometric series, the formula is \(S_n = a \frac{1 - r^n}{1 - r}\).
• \(a\) represents the initial term, \(r\) the common ratio, and \(n\) the total number of terms.
• The formula is particularly useful, as it provides a direct way to compute the sum, avoiding the need to add each term individually.

By leveraging this formula, you can solve problems related to finite geometric sums effortlessly, as demonstrated in the exercise solution. Flexibility and efficiency are its hallmark features, making complex calculations straightforward.