Problem 43 - Algebra for College Students

Question

$$ \sum_{i=1}^{6} 3\left(\frac{1}{2}\right)^{i} \quad 2 \frac{61}{64} $$

Step-by-Step Solution

Answer

The sum is \( 2 \frac{61}{64} \).

Step 1: Understand the Series

The expression given is a summation: \( \sum_{i=1}^{6} 3\left(\frac{1}{2}\right)^{i} \). This is a geometric series where the first term \( a = 3(\frac{1}{2})^{1} = \frac{3}{2} \) and the common ratio \( r = \frac{1}{2} \).

Step 2: Identify 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 \( S_n = a\frac{1-r^n}{1-r} \).

Step 3: Substitute the Known Values into the Formula

Substitute \( a = \frac{3}{2} \), \( r = \frac{1}{2} \), and \( n = 6 \) into the formula: \[ S_6 = \frac{3}{2} \cdot \frac{1-(\frac{1}{2})^6}{1-\frac{1}{2}} \]

Step 4: Calculate \((\frac{1}{2})^6\)

Determine the value of \((\frac{1}{2})^6\): \[ (\frac{1}{2})^6 = \frac{1}{64} \]

Step 5: Compute the Series

Substitute \((\frac{1}{2})^6\) back into the sum formula: \[ S_6 = \frac{3}{2} \cdot \frac{1-\frac{1}{64}}{\frac{1}{2}} \] \[ S_6 = \frac{3}{2} \cdot \frac{\frac{63}{64}}{\frac{1}{2}} \] \[ S_6 = \frac{3}{2} \cdot \frac{63}{32} \] Simplify: \[ S_6 = \frac{189}{64} \]

Step 6: Convert to Mixed Number

Convert \( \frac{189}{64} \) to a mixed number. Divide 189 by 64, the quotient is 2 remainder 61, so: \[ \frac{189}{64} = 2 \frac{61}{64} \]

Step 7: Final Answer

The final solution matches the provided value in the problem. The series sums to \( 2 \frac{61}{64} \).

Key Concepts

Summation Notation

Summation notation is a convenient way to represent the addition of a sequence of numbers, typically following a pattern or rule. In essence, it provides a shorthand for adding up terms that follow a series. This is often symbolized by the Greek letter Sigma (\( \Sigma \)).

In our example, we see this notation in \( \sum_{i=1}^{6} 3\left(\frac{1}{2}\right)^{i} \), which tells us to calculate the sum by iterating the expression \( 3\left(\frac{1}{2}\right)^{i} \) for values of \( i \) from 1 to 6.

• The lower number, 1, is the starting index, indicating where to begin the series.
• The upper number, 6, is the stopping index, marking the final value to use in the calculations.

Using summation notation simplifies complex additions, especially when dealing with many terms or a formulaic pattern in a sequence.

Mastering summation notation allows for easy manipulation of series and mathematical sequences, making them less cumbersome and more systematic.

Sum Formula for Geometric Series

A geometric series is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. To find the sum of the first \( n \) terms in a geometric series, we use the sum formula:

\[ S_n = a\frac{1-r^n}{1-r} \]
Here:

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

In the exercise, the series starts at \( a = \frac{3}{2} \) with a common ratio \( r = \frac{1}{2} \) and we are summing the first 6 terms.

Following the steps, we've plugged these values into the formula, computed the powers, and performed the necessary arithmetic, ending up with a total sum of \( \frac{189}{64} \). Understanding this formula is key for tackling problems involving geometric series efficiently.

Converting Improper Fractions to Mixed Numbers

Converting an improper fraction to a mixed number is a vital skill in simplifying expressions and making calculations more comprehensible. An improper fraction is one where the numerator is larger than the denominator. To convert it:

1. **Divide** the numerator by the denominator.2. The **whole number** result becomes the whole part of the mixed number.3. The **remainder** of the division becomes the new numerator.4. The **denominator** remains the same.
In this exercise, the sum \( \frac{189}{64} \) is an improper fraction. By dividing 189 by 64, we get a quotient of 2 with a remainder of 61. Thus, we convert it to the mixed number \( 2 \frac{61}{64} \).

This process of converting helps in visualizing quantities in a fraction, making them easier to interpret and use. Mastering this concept is helpful in both academic and real-world applications, enhancing numerical clarity and precision.